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ELEMENTARY    TREATISE 


ON 


ALGEBRA, 

THEORETICAL  AND  PRACTICAL, 


ADAITED  TO  tHE  INSTRUCTION  OF  YOUTH  IN  SCHOOLS  AND 
COLLEGES. 


BY  JAMES  RYAN,  A.  M., 

▲vraox  OF  the  differential  and  integral  calcolos  ;  thb  itiw 

AMERICAN  GRAMMAR  OF  ASTRONOMT,  &C. 


TO  WHICH  IS  ADDEO, 

AN   APPENDIX, 

CONTAINING  AN  ALGEBRAIC  METHOD  OF  DEMONSTRATING  THE  PROPOSI- 
TIONS IN  THE  FIFTH  BOOK  OF    EUCLID'S  ELEMENTS,  ACrOHDINO* 
TO  THE  TEXT  AND  ARRANGEMENT  IN  SIMSON'S  EDITION, 

By  ROBERT  ADRAIN,  L.L.D.  F.A.P.S.  F.A.A.S.,  &c 

ANlfPBOFESSOR  OF  MATHEMATICS  AND  NATURAL  PHILOSOPHT,  IN  COLUM- 
BIA COLLEGE,  NEW- YORK. 


SIXTH    EDITION, 

GREATLY    ENLARGED    AND    IMPROVED,    BY   THE    AUTHOR. 

NEW   YORK: 
W.  E.  DEAN,  PRINTER  &  PUBLISHER,  2  ANN  STREET. 


1851. 
OF  THE 

!(    UNIVERSITY 


r^ 


OP  K 


Entkred 

According  to  the  Act  of  Congress  in  the  year  1847,  by 

W.  E.  DEAN. 

In  the  Clerk's  Office  of  the  Southern  District  of 

.    New  York. 


STEREOTYPED  BY  SMITH  &  WRIGHT. 


ADVERTISEMENT 

TO 

THE  FOURTH  EDITION. 


The  author  has  endeavoured  to  accommodate  his  Algebra 
to  the  present  state  of  science  in  the  United  States.  Consider- 
able alterations  and  improvements  have  been  made  in  the  dif- 
ferent sections  of  the  original  work.  There  are  also  intro- 
duced two  new  chapters,  containing  Figurate  and  Polygonal 
Numbers,  Vanishing  Fractions,  Indeterminate  Coefficients,  In- 
determinate and  Diophantine  Analysis.  The  chapters  upon 
these  subjects  have  chiefly  been  derived  from  Euler,  Bonny- 
castle,  Young,  and  Bourdon. 

JAMES  RYAN. 

Neva  York,  July  4, 1838. 


til  89/1' 


ADVERTISEMENT. 


As  Utility  is  the  great  object  aimed  at  in  this  Publication, 
I  have  spared  no  pains  to  make  a  careful  selection  of  materi- 
als, from  the  most  approved  sources,  which  may  tend  to  eluci- 
date, in  a  full  and  clear  manner,  the  Elements  of  Algebra,  both 
in  theory  and  practice. 

Those  authors  of  whose  labours  I  have  principally  availed 
myself,  are  Euler^  Clairaut,  Lacroix,  Garnier^  Bezout,  La' 
grange^  Newton^  Simson^  EmersoUf  Woody  Bonnycastle,  Bridge^ 
and  Bland. 

To  Bland's  Algebraical  Problems,  (a  work  compiled  for  the 
use  of  Students  in  one  of  the  first  Universities  in  Europe),  I 
am  chiefly  indebted  for  the  problems  in  Simple,  Pure,  and 
Quadratic  Equations. 

By  permission  of  the  learned  Dr.  Adrain,  I  have  added,  as  an 
Appendix,  his  method  of  demonstrating  algebraically  the  pro- 
positions in  the  fifth  book  of  Euclid's  Elements. 

JAMES  RYAN. 
New  York,  July  1, 1824. 


CONTENTS.   ^ 


INTRODUCTION. 


VAOI 


Explanation  of  the  Algebraic  method  of  notation  .*— 
Definitions  and  Axioms^  ...........  1 

CHAPTER  I. 

On  the  Addition^  Subtraction^  Multiplication,  and  Division 
of  Algebraic  Quantities. 

SECT. 

I.  Addition  of  algebraic  quantities ---  H 

II.  Subtraction  of  algebraic  quantities       ......  18 

III.  Multiplication  of  algebraic  quantities    .._..-  22 

IV.  Divisionof  algebraic  quantities  ----..-.  33 
V.   Some  general  theorems,  observations,  &c.      -    -    -    -  50 

CHAPTER  II. 
On  Algebraic  Fractions, 

8ECT. 

I.  Theory  of  algebraic  fractions    ---....-  55 
n.  Method  of  finding  the  greatest  common  divisor  of  two 

or  more  quantities 60 

III.  Method  of  finding  the  least  common  multiple  of  two  or 

more  quantities   -----.--...-  72 

rV.  Reduction  of  algebraic  fractions '-.  73 

V.  Addition  and  subtraction  of  algebraic  fractions  -    -    -  85 

VI.  M,ultiplication  and  division  of  algebraic  fractions    .    -  91 

CHAPTER  III. 
On  Simple  Equations  involving  only  one  unknoum  Quantity  95 

SBCT. 

I.  Reduction  of  simple  equations    .-------  96 

H.  Resolution  of  simple  equations,  involving  only  one  un- 
known quantity  -- 101 

ni.  Examples  in  simple  equations,  involving  only  one  un- 
known quantity .---..        108 

1* 


vi  ^B  CONTENTS. 


CHAPTER  IV. 

PAOB 

On  the  Solution  of  Problems  producing  Simple  Equations        117 

SECT. 

I.  Solution  of  problems  producing  simple  equations,  in 

volving  only  one  unknown  quantity     -----        117 

*         CHAPTER  V. 

On  Simple  Equations  involving  two  or  more  unknown 
quantities  --------------        134 

SSCT. 

I.  Elimination  of  unknown  quantities  from  any  number  of 

simple  equations       -----------        134 

II.  Resolution  of  simple  equations,  involving  two  unknown 

quantities  --------------        140 

Examples  in  which  the  preceding  rules  are  applied  in 
the  solution  of  simple  equations,  involving  two  un- 
known quantities     -----------        152 

m.  Resolution  of  simple  equations,  involving  three  or  more 

unknown  quantities  - 158 

IV.  Solution  of  problems  producing  simple  equations,  in- 
volving more  than  one  unknown  quantity  -----        164 

CHAPTER  VI. 

On  the  Involution  and  Evolution  of  Numbers^  and  of 
Algebraic  Quantities. 

SSCT. 

I.  Involution  of  algebraic  quantities    -------  175 

n.  Evolution  of  algebraic  quantities 184 

ni.  Investigation  of  the  rules  for  the  extraction  of  the  square 

and  cube  roots  of  numbers  ---- 191 

CHAPTER  VII. 
On  Irrational  and  Imaginary  Quantities. 

SECT. 

I.  Theory  of  irrational  quantities  --------        1^ 

II.  Reduction  of  radical  quantities  or  surds  -----        203 

III.  Application  of  the  fundamental  rules  of  arithmetic  to 

surd  quantities -----        207 

IV.  Method  of  reducing  a  fraction  whose  denominator  is  a 

simple  or  a  binomial  surd,  to  another  that  shall  have 

rational  denominator ..---        213 

V.  Method   of  extracting   the    square  root  of  binomial 

surds 219 

VI.  Calculation  of  imaginary  quantities     ......        221 


CONTENTS.  ?a 


CHAPTER  VIII. 

PAOK 

On  Pure  Equations     -----------        225 

SECT. 

I.  Solution   of  pure  equations  of  the  first   degree  by 

involution       -------------        225 

II.  Solution  of  pure  equations  of  the  second,  and  other 

higher  degrees,  by  evolution 230 

ni.  Examples  in  which  the  preceding  rules  are  applied 

in  the  solution  of  pure  equations      ------        231 

CHAPTER  IX. 
On  the  Solution  of  Problems  producing  Pure  EquMions        236 

CHAPTER  X. 
On  Quadratic  Equations    -..-....-        242 

SECT. 

I.  Solution  of  adfected  quadratic    equations,   involving 

only  one  unknown  quantity 246 

n.  Solution  of  adfected  quadratic    equations,  involving 

two  unknown  quantities       -      -        254 

CHAPTER  XI. 

On   the    Solution  of  Problems   producing    Quadratic 
•  Equations. 

«ECT. 

I.  Solution  of  problems  producing  quadratic  equations, 

involving  only  one  unknown  quantity  -----        261 
n.  Solution  of  problems  producing  quadratic  equations, 

involving  more  than  one  unknown  quantity    -    -    -        266 

CHAPTER  XII. 
On  the  Expansion  of  Series  .--....-.        272 

SECT. 

I.  Resolution  of  algebraic  fractions,  or  quotients,  into 

infinite  series      ------------        272 

II.  Investigation  of  the  binominal  theorem     -    -     -    -    -        289 

HI.  Application  of  the  binomial  theorem  to  the  expan-     . 

sion  of  series     ---------.._       296 

CHAPTER  XIII. 
On  Proportion  and  Progression. 

SECT. 

I.  Arithmetical  proportion  and  progression  -----        299 
n.  Geometrical  proportion  and  progression    -----        302 


viii.  CONTENTS. 

raCT.  PACK 

III.  Harmonica!  proportion  and  progression     -----        307 

IV.  Problems  in  proportion  and  progression    -    -    -    -    -        308 

CHAPTER  XIV. 
On  Logarithms -----        311 

SECT. 

I.  Theory  of  logarithms      ..........        312 

n.  Application  of  logarithms  to  the  solution  of  exponen- 
tial equations      ............        317 

CHAPTER  XV. 

On  the  Resolution  of  Equations  of  the  third  and  higher 
degrees. 

SECT. 

I.  Theory  and  transformation  of  equations    -    -    -    .    -        320 
II.  Resolution  of  cubic  equations  by  the  rule  of  Cardan, 

or  of  Scipio  Ferreo 326 

in.  Resolution  of  biquadratic  equations  by  the  method  of 

Des  Cartes 331 

IV.  Resolution  of  numeral  equations  by  the  method  of  Di- 
visors ---------------        333 

V.  Resolution  of  numeral  equations,  by  Newton*s  me-   - 

thod  of  approximation    --.-.---.-       336 

CHAPTER  XVI. 

On  Indeterminate  Coefficients,  Vanishing  Fractions^  and  Fig- 

uraie  and  Polygonal  Numbers     -------*   339 

SECT. 

I.  Indeterminate  Coefficients    -----..--        339 

II.  Vanishing  Fractions 342 

in.  Figurate  and  Polygonal  numbers     .-.----        345 

CHAPTER  XVn. 
On  Indeterminate  and  Diophaniine  Analysis  -    -    -    -        347 

SECT. 

I.  Indeterminate  Analysis    -_-.--.---        347 
n.  Diophantine  Analysis  -----------        356 

APPENDIX. 

Algebraic  method  of  demonstrating  the  propositions 
in  the  fifth  book  of  Euclid's  Elements,  according  to 
the  text  and  arrangements  in  Simson's  edition    -    -       87X 


AN 

ELEMENTARY  TREATISE 

ON 

ALGEBRA. 


INTRODUCTION. 

EXPLANATION  OF    THE    ALGEBRAIC    METHOD    OF    NOTATION  I— - 
DEFINITIONS    AND    AXIOMS. 

1.  Algebra  is  aigeneral  method  of  imputation,  in  which 
abstract  quantities  and  their  several  relations  are  made  the 
subject  of  calculation,  by  means  of  ^  alphabetical  lettei*  and 
other  signs. 

2.  The  letters  of  the  alphabet  may  be  employed  at  plea- 
sure for  denoting  any  quantities,  as  algebraical  symbols  or  ab- 
breviations ;  but,  in  general,  quantities  whose  values  are 
known  or  determined,  are  expressed  hyxhe  first  letters,  a,  b,  c, 
&c. ;  and  uj^nown  or  undetermined  quantities  are  denoted  by 
the  last  or  final  ones,  w,  v,  w,  x,  &c. 

3.  Quantities  are  equal  when  they  are  of  the  same  magni-^ 
tude.  The  abbreviation  a  =  b  implies  that  the  quantity  de- 
noted by  a  is  equal  to  the  quantity  denoted  by  b,  and  is  read 
a  equal  to  b ;  a>6,  or  a  greater  than  b,  that  the  quantity  a  is 
greater  than  the  quantity  6;  and  a<^b,  or  a  lesslhan  b^  that 
the  quantity  a  is  less  than  the  quantity  b. 

4.  Addition  is  the  joining  of  magnitudes  into  one  sum.  The 
sign  of  addition  is  an  erect  cross  ;  thus,  a-{-b  implies  the  sum 
of  a  and  b,  and  is  called  a  plus  b,  if  a  represent  8  and  b,  4  ; 
then,  a-\-b  represents  12,  or  a4-^  =  8-|-4  =  12. 

5.  Stibtraction  is  the  taking  as  much  from  one  quantity  as 
is  equal  to  another.  Subtraction  is  denoted  by  a  single  line  ; 
as  a~b,  or  a  iniuiis  b,  which  is  the  part  of  a  remaining,  when 
a  part  equal  to  b  has  been  taken  from  it ;  if  a=:9,  and  b  =  5  ; 
a—b  expresses  9  diminished  by  5,  Which  is  equal  to  4,  or 
a-b=9—5=4. 

2 


2  INTRODUCTION. 

6.  Also^the  difference  of  two  quantities  a  and  h  ;  when  it 
is  not  known  which  of  them  is  the  greater,  is  represented  by 
the  sign -w;  thus,  a-»'6  is  a—h,  ox  h—a\  and  a+h  signifies 
the  sum  or  difference  of  a  and  b.  '*' 

7.  Multiplication  is  the  adding  together  so  many  numbers 
or  quantities  equal  to  the  multiplicand  as  there  are  units  in  the 
multiplier,  into  one  sum  called  the  product.  Multiplication  is 
expressed  by  an  oblique  cross,  by  a  point,  or  by  simple  appo- 
sition ;  thus,  axh^  a  .  b,  or  ab,  signifies  the  quantity  denoted 
by  a,  is  to  be  multiplied  by  the  quantity  denoted  hy  b;  if  «  =  5 
and  b=i7\  then  a  X  5=5  X  7  =  35,  or  a  .  i=-5  .  7  =  35,  or 
ab=5x7  =  35. 

Scholium.  The  multiplication  of  numbers  cannot  be  ex- 
pressed by  simple  apposition.  A  unit  is  a  magnitude  consi- 
dered as  a  whole  complete  within  itself.  And  a  whole  num- 
ber is  composed  of  units  by  continued  additions  ;  thus,  one 
plus  one  composes  two,  2-{-lz=z3,  3  +  1=4,  &c. 

8.  Division  is  the  pbtraction  of  one  quantity  from  another 
as  often  as  it  is  contamed  in  it ;  or  the  finding  of  that  quo- 
tient, which,  when  multiplied  by  a  given  divisor,  produces  a 
given  dividend. 

Division  is  denoted  by  placing  the  dividend  before  the  sign 
-T-,  and  the  divisor  after  it ;  thus  a-^b,  implies  that  the  quan- 
tity a  is  to  be  divided  by  the  quantity  b.  Also,  it  is  frequently 
denoted  by  placing  one  of  the  two  quantities  over  the  other, 

in  the  form  of  a  fraction  ;  thus,  j-  =:  a-^b  ;  if  a^  12,  &  =  4  ; 

then  a-7-J=Y=12-^4=— -— 3. 
b  4 

9.  A  simple  fraction  is  a  number  which  by  continual  addition 
composes  a  unit,  and  the  number  of  such  fractions  contained 
in  a  unit,  is  denoted  by  the  denominator,  or  the  number  below 
the  line  ;  tnus,  j4-g-+^=l-  A  nu?nber  composed  of  such  sim- 
ple fractions,  by  continual  addition,  may  properly  be  termed  a 
multiple  fraction  ;  the  number  of  simple  fractions  composing  it, 
is  denoted  by  the  upper  figure  or  numerator.  In  this  sense, 
^,  f,  ^,  are  multiple  fractions  ;  and  f  =  1,1^=^4-^=1-1— J-=l^. 

10.  When  any  quantities  are  enclosed  in  a  parenih*eses,  or 
have  a  line  drawn  over  them,  they  are  considered  as  one 
quantity  with  respect  to  other  symbols  ;  thus  a— (6-i-c),  or 
a—b-^c\  implies  the  excess  of  a  above  the  sum  of  b  and  c. 
Let  a=9,  &=3,  and  c=2  ;  then  a-(i-f  c)  =  9— (3+2)=9 
— 5=4,  or  a— 6-fc=9— 3+2  =  9— 5=4.      Also,   {a-\-b)x 


INTRODUCTION. 


{c-\-d),  or  a-\-bX  c-{-d,  denotes  that  the  sum  of  a  and  b  is  to  be 
muhiplied  by  the  sum  of  c  and  d  ;  thus,  let  a=:4,  i  =  2,  c  =  3, 
and  (/=5;  then_(a4;^X(c-M)  =  (4  +  2)x(3-f-5)=:6x8=: 
48,  ora+bxc-{-d=4  +  2  x 34-5  =  6  X  8=48.     And  (a—b)-^ 

(c-\-d),  or  ;  implies  the  excess  of  a  alJbve  5,  is  to  be  di- 

vided by  the  sura  of  c  and  d;  if  a=12,  5=2,  c=4,  and  d=l ; 
then,  (a-5)-i-(c+c/)=(12-2)-j-(4-}-l)  =  10^5=2,or  -^ 

_12-2_1Q_ 
~"  4-f  1  "~5  ~   * 

The  line  drawn  over  the  quantities  is  sometimes  called  a 
vinculum. 

11.  Factors  are  the  numbers  or  quantities,  from  the  multi- 
plication of  which,  the  proposi3d  numbers  or  quantities  are 
produced  ;  thus,  the  factors  of  35  are  7  and  5,  because  7x5 
=  35  ;  also,  a  and  b  are  the  factors  o(  ab ;  3,  a^,  b  and  c^,  are 
the  factors  of  Sa^bc'^ ;  and  a-f  5  and  a—b  are  the  factors  of  the 
product  (a-\-b)x{a  —  b). 

When  a  number  or  quantity  is  produced  by  the  multiplica- 
tion of  two  or  more  factors,  it  is  called  a  composite  number 
or  quantity  ;  thus,  35  is  a  composite  number,  being  produced 
by  the  product  of  7  and  5  ;  also,  5acx  is  a  composite  quantity, 
the  factors  of  which  are  5,  a,  c,  and  x. 

12.  When  the  factors  are  all  equal  to  each  other,  the  pro- 
duct is  called  a  power  of  one  of  the  factors,  and  the  factor  is 
called  the  root  of  the  product  or  the  power.  When  there  are 
two  equal  factors,  the  product  is  called  the  second  power  or 
square  of  either  factor,  and  the  factor  is  called  the  second  root 
or  square  root  of  the  power.  When  there  are  three  equal  fac- 
tors, the  product  is  called  the  third  power  or  cube  of  either 
factor,  and  the  factor  is  called  the  third  root  or  cube  root  of  the 
power.     And  so  on  for  any  number  of  equal  factors. 

1 3.  Instead  of  setting  down  in  the  manner  of  other  products, 
the  equal  factors  which  multiplied  together  constitute  a  power, 
it  is  evidently  more  convenient  to  set  down  only  one  of  the 
equal  factors,  (or,  in. other  words,  the  root  of  the  power,) ^nd 
to  designate  their  number  by  small  figures  or  letters  placed 
near  the  root.  These  figures  or  letters  are  always  placed  at 
the  upper  and  right  side  of  the  root,  and  are  called  the  indices 
or  exponents  of  the  power. 

For  example  : 

aX«XaXaor  aaaa  is  denoted  thus,  a* ; 
yXyXyXyxy  or  yyyyy,  thus,  y^\ 


4  INTRODUCTION. 

where  a*  and  y^  are  the  powers  ;  a  and  y  the  roots,  and  4  and 

5  the  indices  or  exponents  of  the  powers.  Again  :  ^ax^  X 
4aa;2x4aa72,  is  thus  abridged,  (4aa;2)3 ;  where  {^ax^^  is  the 
power,  4aa:2  the  root,  and  3  the  index  or  exponent  of  the 
power.  The  same  method  is  adopted,  whatever  be  the  form 
of  the  root:  thus,  (c^ — x^ — y^)x{a^—x^ — y^)x(a^ — x^—y"^) 
is  written  briefly  thus,  (a^ — x'^—y'^Y,  where  (a^—x'^  —  y'^Y  is 
the  power,  a^ — x'^—y'^  the  root  of  the  power,  and  3  its  index 
or  exponent. 

N.  B.  Care  must  always  be  taken,  to  embrace  the  root  in 
parentheses,  except  where  it  is  expressed  by  a  single  charac- 
ter. 

14.  The  coefficient  of  a  quantity  is  the  number  or  letter  pre- 
fixed to  it ;  being  that  which  shows  how  often  the  quantity  is 
to  be  taken-;  thus,  in  the  quantities  Zh  and  Sx^^  3  and  5  are 
the  coefficients  of  h  and  x"^.  .  Also,  in  the  quantities  3«y  and 
ba?x~  3a  and  ba^  are  the  coefficients  of  y  and  x. 

15.  When  a  quantity  has  no  number  prefixed  to  it,  the 
quantity  has  unity  for  its  coefficient,  or  it  is  supposed  to  be 
taken  only  once  ;  thus,  x  is  the  same  as  Ijr;  and  when  a 
quantity  has  no  sign  before  it,  the  sign  +  is  always  under- 
stood;  thus,  Za^b  is  the  same  as  -\-'da%,  and  5a— 36  is  the 
same  as -f  5a— 36.     . 

16.  Quantities  which  can  be  expressed  in  finite  terms,  or 
the  roots  of  which  can  be  accurately  expressed,  are  rational 
quantities  ;  thus,  3a,  Ja,  and  the  square  root  of  Aa^,  are  ra- 
tional quantities  ;  for  if  a=10  ;  then,  3a=:3  X  10  =  30  ;  fa  = 
|-X  10:::=2_o_4  .  ^^^  (\^q  squarc  root  of  Aa^=:.  the  square  root 
of  4  X  102=  thg  square  root  of  4x10x10=  the  square  root 
of  400  =  20. 

17.  An  irrational  quantity,  or  surd,  is  that  of  which  the 
value  cannot  be  accurately  expressed  in  numbers,  as  the 
square  root  of  3,  5,  7,  &c.  ;  the  cube  root  of  7,  9,  &c. 

18.  The  roots  of  quantities  are  expressed  by  means  of  the 
radical  sign  y,  with  the  proper  index  annexed,  or  by  fraction- 
al indices  placed  at  the  right-hand  of  the  quantity  ;  thus,  -y/a, 

1  1 

or  a^,  expresses  the  square  root  of  a  ;  ^  (a  +  a:),  or  (a-fa;)  , 

the  cube  root  of  {a-\-x)\  ^(a-f-x),  or  (a+a?)"*,  the  fourth 
root  of  (a-\-x).     When  the  roots  of  quantities  are  expressed 

by  fractional  indices  ;  thus,  a  ,  (a-\-x)^,  (a-f-a?)'*  ;  they  are 
generally  read  a  in  the  power  (J),  or  a  with  (J)  for  an  index ; 
(a-\-x)  in  the  power  (^),  or  {a-y-x)  with  (J)  for  an  index;  and 
(a-\-x)  in  the  power  (i),  or  {a-\-x)  with  (\)  for  an  index. 

19.  Like  quantities  are  such  as  consist  of  the  same  letters  or 


INTRODUCTION.  5 

the  same  combination  of  letters,  or  that  differ  only  in  their 
numeral  coefficients ;  thus,  5a  and  7a  ;  4ax  and  9ax ;  +2ao 
and  9ac ;  — 5ca  ;  &c.,  are  called  like  quantities  ;  and  unlike 
quantities  are  such  as  consis*t  of  different  letters,  or  of  diffe't- 
ent  combination  of  letters  ;  thus,  4a,  3&,  7ax,  daj/"^,  &c.  are 
unlike  quantities. 

20.  Algebraic  quantities  have  also  different  denominations, 
according  to  the  sign  -j-,  or — . 

Positive,  or  affirmative  quantities,  are  those  that  are  addi- 
tive, or  such  as  have  the  sign  +  prefixed  to  them ;  as,  -\-a, 
-^dab,  or  9ax. 

21.  Negative  quantities  are  those  that  are  subtractivc,  or 
such  as  have  the  sign  —  prefixed  to  them  ;  as,  — x^  — 3a^, 
—  4ab,  &c.  A  negative  quantity  is  of  an  opposite  nature  to  a 
positive  one,  with  respect  to  addition  and  subtraction  :  the 
condition  of  its  determination  being  such,  that  it  must  be  sub- 
tracted when  a  positive  quantity  would  be  added,  and  the  re- 
verse. 

22.  Also  quantities  have  dif&rent  denominations,  according 
to  the  number  of  terms  (connected  by  the  signs  -f-  or  — )  of 
which  they  consist ;  thus,  a,  3b,  —4ad,  &c.,  quantities  con- 
sisting of  one  term,  are  called  simple  quantities,  or  monomi- 
als ;  a-^-x,  a  quantity  consisting  of  two  terms,  a^jjMnomial ; 
a~x  is  sometimes  called  a  residual  quantity.  A  trinomial  is 
a  quantity  consisting  of  three  terms  ;  as,  a-{-2x — Sy  ;  a  quad- 
rinomial  o(  four ;  as,  a-^b-{-3x  —  4y ;  and  a  polynomial,  or 
multinomial,  consists  of  an  indefinite  number  of  terms.  Quan- 
tities consisting  of  more  than  one  term  may  be  called  compound 
quantities. 

23.  Quantities  the  signs  of  which  are  all  positive  or  all 
negative,  are  said  to  have  like  signs  ;  thus,  -{-3a,  4-4a?, 
+  5a&,  have  like  signs  ;  also,  —4a,  —3b,  —4ac.  When  some* 
are  positive,  and  others  negative,  they  have  unlike  signs ; 
thus,  the  quantities  +3a  and  — 5ab  have  unlike  signs  ;  also, 
the  quantities  '-3ax,  -\-3a'^x  :  and  the  quantities  —b,  -j-b. 

24.  If  the  quotients  of  two  pairs  of  numbers  are  equal,  the 
numbers  are  proportional,  and  the  first  is  to  the  second,  as  the 
third  to  the  fourth  ;   and  any  quantities,  expressed  by  such 

d       c 
numbers,  are   also  proportional;  thus,  if -7-=-^;  then  a  is  to 

5  as  c  to  d.     The  abbreviation  of  the  proportion  ;  a  :  b  i:  c  : 

d  ;  and  it  is  sometimes  written  a  :  b=c  :  d;  if  a  =8,  6=4 

8       12 

c=12,  and  d  =6 ;  then,  -=  -^-=2,  and  8  :  4  ::  12  ;  6. 

4       o 


6  INTRODUCTION. 

25.  A  term,  is  any  part  or  member  of  a  compound  quanti- 
•ty,  which  is  separated  from  the  rest  by  the  signs  -|-  and  — , 

thus,  a  and  b  are  the  terms  of  a-f-^  ;  and  3a,  —'Zb,  and  -fSac?, 
are  the  terms  of  the  compound  quantity  3a — 2b  -{-bad.  In 
like  manner,  the  terms  of  a  product,  fraction,  or  proportion, 
are  the  several  parts  or  quantities  of  which  they  are  compos- 
ed ;  J;hus,  a  and  b  are  the  terms  of  ab^  or  of  ^ ;  and  a,  5,  c,  <f, 

b 
are  the  terms  of  the  proportion  a  :  b  ",*,  c:  d. 

26.  A  measure^  or  divisor^  of  any  quantity,  is  that  which  is 

contained  in  it  some  exact  number  of  times  ;    thus,  4  is  a 

35a 
measuje  of  12,  and  7  is  a  measure  of  35a,  because  ---=5a. 

27.  A  prime  number,  is  that  which  has  no  exact  divisor, 
except  itself,  or  unity ;  2,  3,  5,  7,  11,  &c.  and  the  interven- 
ing numbers  ;  4,  6,  8,  &c.  are  composite  numbers.     (Art.  12.) 

28.  Commensurable  numbers,  or  quantities,  are  such  as 
have  a  common  measure  ;  thus,  6  and  8,  8aa;,  and  4J,  are 
commensurable  quantities ;  th*e  common  divisors  being  2  and 
4  ;  also,  Aax^  and  bax  are  commensurable,  the  common  divi- 
sor being  ax. 

29.  Ako,  two  or  more  numbers  are  said  to  be  prime  to 
each  otiBr,  when  they  have  no  common  measure  or  divisor, 
except  unity  ;  as  3  and  5,  7  and  9,  11  and  13,  &c. 

30.  A.  .multiple  of  any  quantity,  is  that  which  is  some  ex- 
act number  of  times  that  quantity  ;  thus,  12  is  a  multiple  of 

15a 
4  ;  and  15a  is  a  multiple  of  3a,  because  ~— =5. 

3a 

31.  The  reciprocal  of  a  quantity  is  that  quantity  inverted  or 

unity  divided  by  it.     Thus,  the  reciprocal  of  a,  or  of  -  is  -,  the 

reciprocal  of  7-  is  -  and  the  reciprocal  of  — —r  is r. 

^  b     a  a-\-b      a — 0 

32.  The  reciprocal  of  the  powers  and  roots  of  quantities, 
is  frequently  written  with   a  negative    index  or  exponent ; 

thus,  the  reciprocal  of  a^=—^  maybe  written  a    ^;  the  re- 
ciprocal of  (a-\-xY=' — ; — TT,  may  be  written,  (a-\-x)     3;  but 
'^  \         /•      [a-^xy 

this  method  of  notation  requires  some  farther  explanation, 
which  will  be  given  in  a  subsequent  part  of  the  work. 

33.  A  function  of  one  or  more  quantities,  is  an  expression 
into  which  those  quantities  enter  in  any  manner  whatever, 


INTRODUCTION. 


either  combined,  or  not,  with  known  quantities  ;  thus,  <Zr|-2a7, 

1. 
ax-\-3ax^,  5ax^ — Sa^,  &c.  are  functions  of  a;;  and  3ax'^-\-xy'^ 
2  {x'^-\-5xi/)^,  &c.  are  functions  of  a:  and  y. 

34.  When  quantities  are  connected  by  the  sign  of  equali- 
ty, the  expression  itself  is  called  an  equation;  thus,  a-{-b=c 
-\-d,  means  that  the  quantities  a  and  b,  are  equal  to  the  quan- 
tities c  and  d ;  and  this  is  called  an  equation  ;  it  is  divided  into 
two  members  by  the  sign  of  equality,  a-{-b  is  the  firstj  and 
c-{-d  the  second  member  of  the  equation. 

35.  In  algebraical  operations  the  word  therefore,  or  conse- 
quently, often  occurs.  To  express  this  word,  the  sign  .-.  is 
generally  made  use  of:  thus,  d—h,  therefore,  a-{-cz=.b-\'C\  is 
expressed  .-.  a-\-cz=.b-\-c. 

Also  00  is  the  sign  of  infinity  ;  signifying  that  the  quantity 
standing  before  it  is  of  an  unlimited  value,  or  greator  than  any 
quantity  that  can  be  assigned. 

36.  The  signs  +  and  — ,  give  a  kind  o^  quality  or  affection 
to  the  quantities  to  which  they  are  annexed.  As  all  those 
terms  which  have  the  sign  -|-  prefixed  to  them,  are  to  be 
added  (Art.  4),  and  those  quantities  which  have  the  sign  — 
prefixed  to  them,  are  to  be  subtracted,  (Art.  5),  from  the  terms 
which  precede  them  ;  the  former  has  a  tendency  to  increase, 
and  the  latter  to  diminish,  the  quantities  with  which  they  are 
combined  ;  thus,  the  compound  quantity,  a — x,  will  therefore 
be  positive  or  negative,  according  to  the  effect  which  it  pro- 
duces upon  some  third  quantity  b;  if  a  be  greater  than  x,  then, 
(since  a  is  added,  and  b  subtracted)  b-\ra — a;  is  '^b  ;  but  if  a  be, 
less  than  x  ;  then,  b-\-a — x  is  <^b. 

In  the  first  place,  let  a=10,  a;=:z6,*and  b  —  Q  ;  then  b-\-a^x 
=  8-1-10  —  6,  which  is  >8  ;  since  10—6=4,  a  positive  quan- 
tity ;  therefore,  a — x  is  positive.  Next,  let  a=\2,  a:=14,  and 
b—20\  then  b-\-a—x— 20 -{-12 — 14,  which  is  <20  ;  since 
12 — 14  =  — 2,  a  negative  quantity  ;  therefore  a — x  is  negative. 
In  like  manner,  it  may  be  shown  that  the  expression  a — b-\-c 
— d  is  positive  or  negative  according  as  a-\-c  is  >  or  <^b-\-d\ 
and  so  of  all  compound  quantities  whatever. 

37.  The  use  of  these  several  signs,  symbols,  and  abbrevia- 
tions, may  be  exemplified  in  the  following  manner : 


EXAMPLES. 


Example.  1.  In  the  algebraic  expression  c-f-ft-j-c— •(?,  let 
a=8,  5=7,  c=4,  and  d=.Q  ;  then 

a+&+c— £?=8  +  7-f-4— 6  =  19— 6  =  13. 
Ex,  2.    In  the  expression  ab-^-ax — by,  let   a=5,   J=4 


8  INTRODUCTION. 

a: =8,  and  y=:12  ;  then,  to  find  its  value,  we  have  ab-{-aac-^ 
^=5X4  +  5X8—4X12 

=20+40 — 48 

=  60—48  =  12. 

Ex.  3.  What  is  the  value  of -^—^jwhere  a=4,  x=5,  y 

a-f-o  ^ 

nrlO,  and  b=6  1 

Here  3aa:+2y=3x4x 5+2x10=60+20=80,  and  a+ 
fc=4+6=10; 

3aa;+2y__80_ 
■**     a+b     ~10~   ' 
Ex.  4.  What  is  the  value  of  a'^+2ab-'C+dy  when  a=6, 
i=5,  c=4,  and  £^=1  ?     Ans.  93. 

Ex.  5.  What  is  the  value  of  ab-\-ce—bd,  when  a=8,  i=7, 
c=6,  d=5y  and  e  =  l  ?     Ans.  27. 

dx-i—bij 
Ex.  6.  In  the  expression  -j— — ^,  let  a=5,  5=3,  x=7, 

o-{-x 

and  y=5  ;  what  is  its  numerical  value  ?     Ans.  5. 

ax^-\-b^ 
Ex.  7.  In  the  expression  ^^ ,  let  flt=3,  5=5,  c=2, 

6a: — a'' — c 

x=6  ;  What  is  its  numerical  value  ?     Ans.  7. 

Ex.  8.  What  is  the  value  of  a2x(a+6)—2a6c,  where  a=6, 
5  =  5,  and  c=4  ?     Ans.  156. 

Ex.  9.  There  is  a  certain"  algebraic  expression  consisting 
of  three  terms  connected  together  by  the  sign  plus ;  the  first 
term  of  it  arises  from  multiplying  three  times  the  square  of  a 
by  the  quantity  b  ;  the  second  is  the  product  of  a,  b  and  c  ;  and 
the  third  is  two  thirds  of  the  product  of  a  and  b.  Required 
the  expression  in  algebraic  writing,  and  its  numerical  value, 
where  a=4,  5=3,  and  c=2  ?     Ans.  176. 

DEFINITIONS. 

38.  A  proposition,  is  some  truth  advanced,  which  is  to  be  de- 
monstrated, or  proved ;  or  something  proposed  to  be  done  or 
performed ;  and  is  either  a  problem  or  theorem. 

39.  A.  problem,  is  a  proposition  or  question,  stated,  in  order  to 
the  investigation  of  some  unknown  truth ;  and  which  requires 
the  truth  of  the  discovery  to  be  demonstrated. 

40.  A  theorem,  is  a  proposition,  wherein  something  is  advanc- 
ed or  asserted,  the  truth  of  which  is  proposed  to  be  demon- 
strated or  proved. 

41.  K  corollary,  ox  consectary^  is  a  truth  derived  from  some 


INTRODUCTION.  9 

proposition  already  demonstrated,  without  the  aid  of  any  other 
proposition. 

42.  A  lemma,  signifies  a  proposition  previously  laid  down, 
in  order  to  render  more  easy  the  demonstration  of  some  theo- 
rem, or  the  solution  of  some  problem  that  is  to  follow. 

43.  A  scholium,  is  a  note,  or  remark,  occasionally  made  on 
some  preceding  proposition,  either  to  show  how  it  might  be 
otherwise  effected  ;  or  to  point  out  its  application  and  use. 

44.  An  axiom,  is  a  self-evident  truth,  or  proposition  univer- 
sally assented  to,  or  which  requires  no  formal  proof. 

45.  As  axioms  are  the  first  principles  upon  which  all  ma- 
thematical demonstrations  are  founded,  I  will  point  out  those 
that  are  necessary  to  be  observed  in  the  study  of  Algebra,  as 
there  will  be  frequent  occasion  to  advert  to  them. 

AXIOMS. 

46.  When  no  difference  can  be  shown  or  imagined  between 
two  quantities,  they  are  equal. 

47.  Quantities  equal  to  the  same  quantity,  are  equal  to  each 
other. 

48.  If  to  equal  quantities  equal  quantities  be  added,  the 
wholes  will  be  equal.  Thus,  if  a=zh,  then  a-^-c—b-^-c  ;  if 
a — hz=.c,  then  adding  h,  a—h-\-h=zc-\-h,  or  a=:c-^h. 

49.*If  from  equal  quantities  equal  quantities  be  subtracted, 
the  remainders  will  be  equal. 

If  a  — 6,  then,  a—2=b—2  ;  if  Z>-4-c  =  a-{-c,  then  h=:a. 

50."  If  equal  quantities  be  multiplied  by  equal  numbers  or 
quantities,  the  products  will  be  equal. 

Thus,  if  a=i&,  3a=35;  if  a=:-,  3a=6;    if  a=iJ,  ca=c5  ; 

o 

and  if  a=zh,  a  X  a=h  X  h,  or  a'^=zh'^. 

51 .  If  equal  quantities  be  divided  by  equal  numbers  or  quan- 
tities, the  quotients  will  be  equal. 

Ihus,  II  5a=106, -— =:— :r-,  or  a=2o  :  if  ca=:co. — =z     , 
5        o  c        c ' 

or  az=zb  ;  and  if  c^^=.ba,  then  — = — ,  or  a^=z.b. 

a        a 

Scholium.  Articles  (49),  (50),  (51),  might  have  been  de- 
duced from  Art.  (48) ;  but  they  are  all  easily  admitted  as 
axioms. 

52.  If  the  same  quantity  be  added  to  and  subtracted  from 
another,  the  value  of  the  latter  will  not  be  altered.  Thus,  if 
a=;c,  then  a-\-b=^c-^b^  and  a  +  6  — Z>=c-}-& — ^,  or^a=c. 


10  INTROrrCTION. 

This  might  be  inferred  from  Art.  (48). 

53.  If  a  quantity  be  both  multiplied  and  divided  by  another 

its  value  will  not  be  altered.     Thus,  ifa=6;  then  3a =36 

,,..,.      ^     ^   3a     3b 
and  dividmg  by  3,  — =— ,  or  a=o. 


CHAPTER  I. 

ON  THE 

ADDITIOI^UBTRACTION,  MULTIPLICATION, 

AND 

DIVISION  OF  ALGEBRAIC  QUANTITIES. 


^  1.  Addition  of  Algebraic  Quantities. 

54.  The  addition  of  algebraic  quantities  is  performed  by- 
connecting  those  that  are  unlike  with  their  proper  signs,  and 
collecting  those  that  are  like  into  one  sum;  for  the  more 
ready  effecting  of  which,  it  may  not  be  improper  to  premise 
a  (ew propositions,  from  which  all  the  necessary  rules  may  be 
derived. 

55.  If  two  or  more  quantities  are  like^  and  have  like  signs j  the 
sum  of  their  coeffi,cents  prefixed  to  the  same  letter,  or  letters, 
with  the  same  sign,  will  express  the  sum  of  these  quantities. 
Thus,  5a  added  to  7a  is  =  12a  ; 
And --5a  added  to  —  Sa  is  =  — 8a. 
For,  if  the  symbol  a  be  made  to  represent  any  quantity  or 
thing,  which  is  the  object  of  calculation,  5a  will  represent 
five  times  that  thing,  and  7a  seven  times  the  same  thing,  what- 
ever may  be  the  denomination  or  numeral  value  of  a  ;  and 
consequently,  if  the  quantities  5a  and  7a  are.  to  be  incorpo- 
rated, or  added  together,  their  sum  will  be  twelve  times  the 
thing  denoted  by  a,  or  12a. 

Moreover,  since  a  negative  quantity  is  denoted  by  the  sign 
of  sulJtraction  :  thus,  if  a+&  =  a—c,  b=—c,  and  c=--i.  A 
debt  is  a  negative  kind  of  property,  a  loss  a  negative  gain,  and 
a  gain  a  negative  loss. 

Therefore  it  is  plain  that  the  quantities,— 5a  and— 3a 
will  produce,  in  any  mixed  operation,  a  contrary  effect  to  that 
of  the  positive  quantities  with  which  they  are  conne>ted  ; 
and  consequently,  after  incorporating  them  in  the  same  man- 
ner as  the  latter,  the  sign  —  must  be  prefixed  to  the  result , 
so  that  if  A  be  greater  than  a,  it  is  evident  that  5  (a— a)  -f- 
3(a— a),  or  (5a— 5a)  +  (3A— 3a)r=:8A— 8a  ;  and  therefore  the 
sum  of  the  quantities  — 5a  and — 3a,  when  taken  in  their  iso-^ 
lated  state,  will,  by  a  necessary  extension  of  the  proposition 
be  =— 8a. 


12  ADDITION. 

56.  If  two  quantities  are  like,  hut  have  unlike  signs,  the  difference 

of  their  coefficients,  prefixed  to  the  same  letter,  or  letters,  with 

the  sign  of^tliat  which  hath  the  greater  coeffcient,  will  express 

the  sum  of  those  quantities. 

Thus  -\-6a  added  to— 4a  is  =  4-2<ii^ 
And  — 6a  added  to  -\-4a  is— — 2a. 

Since,  Art.  (36),  the  compound  quantity  a—b-{-c—d,  &c. 
is  positive  or  negative,  according  as  the  sum  of  the  positive 
terms  is  greater  or  less  than  the  sum  of  the  negative  ones,  the 
aggregate  or  sum  of  the  quantities  4a — 2a + 2a— 2a,  or  6a— 4a, 
will  be  4-2a:  since  the  sum  of  the  positive  terms  is  greater 
than  the  sum  of  the  negative  ones.  And  the  sum  of  the  quan- 
tities a — 4a-|-3a — 2a,  or  4a — 6a,  will  be — 2a  ;  since  the  sum 
of  the  negative  terms  is  greater  than  the  sum  of  the  positive 
ones. 

Corollary.  Hence  it  appears,  that  if  the  sum  of  the  posi- 
tive terms  be  equal  to  the  sum  of  the  negative  ones,  their  ag- 
gregate or  sum  will  be  nothing.  Thus  5a— 5a=0  ;  and  5a 
—  3a-|-4a — 6a=:9a — 9a  =  0, 

57.  The  preceding  proposition  is  demonstrated  in  the  fol- 
lowing manner  by  Bonnycastle  in  his  Algebra.  Vol.  II. 
8vo. 

Where  the  quantities  are  supposed  to  be  like,  but  to  have 
unlike  signs,  the  reason  of  the  operation  will  readily  appear, 
from  considering,  that  the  addition  of  algebraic  quantities, 
taken  in  a  general  sense,  or  without  any  regard  to  their  par- 
ticular values,  means  only  the  uniting  of  them  together,  by 
means  of  the  arithmetical  operations  denoted  by  the  signs  -\- 
and  —  ;  and  as  these  are  of  contrary,  or  opposite  natures, 
the  less  quantity  must  be  taken  from  the  greater,  in  order  to 
obtain  the  incorporated  mass,  and  the  sign  of  the  greater  pre- 
fixed to  the  result.  So  that  if  6a  is  to  be  added  to  4  (a— a),  or 
to  4a --4a,  the  result  will  evidently  be  4 a -f- 6a— 4a,  01^4 a 4- 
2a ;  and  if  4a  is  to  be  added  to  6  (a— a),  or  to  6a  — 6a,  the 
result  will  be  6A-h4a — 6a,  or  6a— 2a  ;  wljence,  by  making  this 
proposition  general,  as  in  the  last,  the  sum  of  the  isolated  quan- 
tities 6a  and  —4a  will  be  +2a,  and  that  of  4a  and  —6a  will 
be  —2a. 

58.  If  two  or  more  quantities  be  unlike,  their  sum  can  only  be 
expressed  by  writing  them  after  each  other,  with  their  proper 
signs. 

Thus,  the  sura  of  2a  and  2b,  can  only  be  expressed,  with 
the  sign  -f  between  them,  which  denotes  that  the  operation  of 
addition  is  to  be  performed  when  we  assign  values  to  a  and  b. 


ADDITION.  13 

For,  if  fl5=10,  and  5=5  ;  then  the  sum  of  2a  and  2b  can  be 
neither  4a  nor  4b,  that  is,  neither  4  X  10=40,  nor  4  X  5=20  ; 
but  2x10+2x5=20  +  10  =  30.  In  like  manner,  the  sum  of 
3a,  —5b,  2c,  and  — 8c?,  can  no  otherwise  be  incorporated,  or 
added  together,  than  by  means  of  the  signs  +  and  —  ;  thus, 
3a— 5b -{-2c— 8d. 

These  propositions  being  well  understood,  the  following 
practical  rules,  for  performirig  the  addition  of  algebraic  quan- 
tities, which  is  generally  divided  into  three  cases,  are  readily 
deduced  from  them. 

CASE  I. 

When  the  quantities  are  like,  and  have  like  signs. 


59.  Add  all  the  numeral  coefficients  together,  to  their  sum 
prefix  the  common  sign  when  necessary,  and  subjoin  the 
common  quantities,  or  letters. 

EXAMPLE  1. 

2x-{-3a—4b 
3j;+4a—  b 
7x-\-   a— 7b 

x+9a—9b 
9j:+  a —   b 

x-\-8a  —  3b 


23j;+26a  — 255 

In  this  example,  in  adding  up  the  first  column,  we  say,  1  + 
9+1+7  +  3  +  2=23,  to  which  the  common  letter  x  is  sub- 
joined. It  is  not  necessary  to  prefix  the  sign  +  to  the  result, 
since  the  sign  of  the  leading  term  of  any  compound  algebraic 
expression,  when  it  is  positive,  is  seldom  expressed  ;  for  (14) 
when  a  quantity  has  no  sig-n  before  it,  the  sign  +  is  always 
understood.  And  it  may  be  observed  when  it  has  no  numeral 
coefficient,  unity  or  1  is  always  understood. 

Also,  the  sum  of  the  second  column  is  found  thus,  8+1+9 
+  1  +  4  +  3=26,  to  which  the  sign  +  is  prefixed,  and  the 
common  letter  a  annexed. 

^Again,  the  sum  of  the  third  column  is  found  thus  ;  3-|-l  + 
6+7+1+4=25,  to  which  the  sign  —  is  prefixed,  and  the 

3 


14  ADDITION. 

common  letter  h  subjoined.  So  that  the  sum  of  all  the  quan- 
tities is  expressed  by  23  times  x  plus  26  times  a  minus  25 
times  h.  ♦ 


Ex.2. 

9xy—4hc-\-'7x'^ 
4xy—  bc-\'3x^ 

xy—7bc-\-4x'2 
8xy-'4bc-{'  a:2 
7a:y—  bc-\-9x^ 

xy  —  3bc-{-   a;2 

Ex.  3. 

5^3— 3a:2+3y-.19 

4a3--  x^-\-4y-'17 

a3— 7a;2+7y— 14 

7a^-.   a;2+   y—    1 

8a^—9x'^-\-9y—20 
7«3_lla;2+y—    8 

30xy'-20bc-\-25x^ 

32a3_32a:2-f.25y-79 

Ex.  4.  Add  together  2a;+3a,  4a:-j-a,  5a;-f  8«,  7a: 4- 2a,  and 
x+a.  Ans.  19a;+15a!. 

Ex.  5.  Add  together  7x^  —  5bc,  3x^—bc,  x^—4bc,  5x'^—bc^ 
and  4a;2— 4Z>c.  Ans.  20a;2— I5Z>c. 

Ex  6.  Required  the  sum  of  3x^-{-4x'^—x,  2x^-^x'^~3xj 
7a;3-|-2a;2— 2a:,  and4ar"*+2x2  — 3a:.  Ans.   1 6a:H  9a:2— 9a:. 

Ex.  7.  What  is  the  sum  of  7a^  —  3a^-{-2aP  —  3b^,  ah'^  — 
an  —  b'^-^Aa?,^bh'^^bab'^~4a'^b-{-Q>d\  and  —aF'b~\-4ab'^-^ 
453+a3?  Ans.   \8a^-9a^b+\2ab'^—l3¥. 

Ex.  8.  Add  together  2a;2y— a:+2,  a;2y— 4a:+3,  4x2y— 3af 
+  1,  and  5a;2y_7a;+7.  Ans.  12a:2y-15a:4-13. 

1  I 

Ex.  9.  Required  the  sum  of  30— 13a:^— 3a?y,  23~10a;^— 
4acy,  —  14a:^— 7a:y4-14,  —  5a;y4-10  — 16a;^,and  1— 2a;^~a:y. 

Ans.  78  — 55a;^— 20a'y. 
Ex.   10.    Add  3(a:  +  yf  -  4{a—b)\  {x  +  yf  -  {a-b)\ 
— 7(a— 6)3-f.5(a;+y)2,  and  2(x+yf-(a—by  together. 

Ans.  ll(a:+y)2  — 13(a-6)3. 

CASE  II. 

When  the  quantities  are  like,  but  have  unlike  signs. 

RULE. 

• 
60.  Add  all  the  positive  coefficients  into  one  sum,  and  those 
that  are  negative  into  another  ;  subtract  the  lesser  of  these 
Bums  from  the  greater ;  to  this  difference,  annex  the  common 
letter  or  letters,  prefixing  the  sign  of  the  greater^  and  the  re- 
sult will  be  the  sum  required. 


ADDITION.  15 

EXAMPLE  1, 

7a:3— 3a:2+3a; 
— 4x^-\-  x^ — Ax 
—  c^—2x^+7x 

9x^+6x'2—9x 

3x^-5x^+6x 
— 5x^-\-3x^ — 6a: 


9a;3     *    —3a; 


In  adding  up  the  first  column,  we  say  34-9-}-7  =+  19» 
and  ^(5  +  1+4)  =  — 10;  then,  +19  — 10=+9=  the  ag- 
gregate sum  of  the  coefficients,  to  which  the  common  quantity 
x^  is  annexed. 

In  the  second  column,  the  sum  of  the  positive  coefficients 
is  3  +  64-1  =  10,  and  the  sum  of  the  negative  ones  is  —(5+2 
+  3)  =  —  10;  then,  10 — 10  =  0;  consequently,  (by  Cor.  Art. 
56),  the  aggregate  sum  of  the  second  column  is  nothing.  And 
in  the  third  column,  the  sum  of  the  positive  coefficients  is 
6  +  7  +  3  =  16,  and  the  sum  of  the  negative  one  is — (6  +  9+ 
4)  =  — 19  ;  then  +16  — 19  =  — 3  ;  to  which  the  common  let- 
ter is  annexed. 

Ex.  2  Ex.  3. 

dx"^ — 6a  +  4a:  — 3  4,ab-\-3xi/—2ax-{-  c 

— 2x^'\-   a — 9a;+7  —  ab —  x7/-\-  ax— 5c 

7a:2+7a+7a;  — 1  5ab—2xy—7ax-i-7c 

—  x^  —  3a—2x-i-3  — 4ai+   a:y+   ax-^-  c 

+  3a;2+   a— 4a;+4  'iab—3xy+4.ax—  c 

—  7a;2  — 4a+3a;— 5  —  ab —  xy —  <za:+4c 


5a;'^— 4a—  x-\-b  lOab — 3xy—4ax-j-7c 


Ex.  4. 
3(a  +  6)2-  5(a;2+y2)3_^3(a3^c2)3-f  9a^y 
—  (a+5)^+     (x''-{-y^)^~5{a3+c^Y-4xy 
+  8{a-\-bf~  6(x^+y^y-^8(a^+c^y+  xy 

— 2(a+&)2-      (a-2  +  y2)2_7(^3_|_c2)3_3a.y 

+  5(«+5p-  7(a:2+y2)2_   (a3^c2)3-.   xy 
13(a+5)^— 18(a:2  4.y2)2_2(a3+c2)3+2a;y 


16  ADDITION. 

Ex.  5.  Required  the  sum  of  Aa^y  —5a^,  a^,  —6a^,  9a^,  and 
— a^  Ans.  2a2. 

Ex.  6.  Required  the  sum  of  4a;2— 3a?-f  4,  a:— 2a:2— 5^  1-f. 
3x2_5a:,  2j:— 4  +  7rr2,  13— a;^— 4:^  Ans.  lla;2— 9a;  +  9. 

Ex.  7.  Required  the  sum  of  40?^— 2a;+y,  4x  —  'i/  —  x^,9y-{- 
7x^—x,  21a;— 2y-l-9a;=^.  Ans.  19a;3+22ir+7y. 

Ex.  8.  Required  the  sum  of  5a^—2ab-^b^,  ab—2b^—a\ 
h^  —  3ab  +  4a\4ab-^2a^—4b^.  Ans.  \0a^—4b'\ 

Ex.  9.  What  is  the  sum  of  2a— 3a;2,  5a;2— 7a,— 3aH-a;2, 
and  a  — 3a;2?  *  Ans.  —7a. 

Ex.  ]0.  What  is  the  sum  of  4  — 3a;,  ar— 5,  2a;— 4,  — 4a;-f 
13,  and  -5a;+l  ?  Ans.  9-9ar. 

CASE  III. 

When  the  quantities  are  unlike^  or  when  like  and  unlike 
are  mixed  together. 

"rule. 

61.  When  the  quantities  are  unlike,  write  them  down,  one 
after  another,  with  their  signs  and  coefficients  prefixed ;  but 
when  some  are  like,  and  others  unlike,  collect  all  the  like 
quantities  together,  by  taking  their  sums  or  differences,  as  in 
the  foregoing  cases,  and  set  down  those  that  are  unlike  as 
before. 

Example  1.  Add  together  the  quantities  7a2,  —56,  -\-4d, 
—  9a,  and  Sc^. 

Here,  the  quantities  are  all  unlike ;  .•.  (Art.  58),  their  sum 
must  be  written  thus  ; 

7a2— 5J+4ti— 9a4  8c2. 
When  several  quantities  are  to  be  added  together,  in  what- 
ever order  they  are  placed,  their  values  remain  the  same. 
Thus,  7a2_564-4(i— 9a-h8c2,  8c2— 55  + 4rf-9a  +  7a2,  or 
4d — 56— 9a-f-8c2-|-7a2,  are  equivalent  expressions:  though 
it  is  usual,  in  such  cases,  to  take  them  so  that  the  leading  term 
s^ll  be  positive. 

Ex.2. 

3a;—     y+   d 

4a—     x—By 

5xy-\-7ax-{-  y^ 

Zax — 2a;y  +  4a;^ 

5y4-  2c/4-5ar 


7a;-j_y4.3fi4.3a;y+10aa:+4a-f-y2_^4a;2 


ADDITION.  17 

Ex.  3. 

4a;3— 3a:y+3y  —3    — Sa:^ 

30  +6a;2+2a?  —3y'^—2x^ 
2x^—8     —bxy—ly  — 2y2 


7a:3 — Say + y  + 1 9  +  2a;2 + 2ar. 


In  Ex.  2.  Collecting  together  like  quantities,  and  beginning 
with  3a7,  we  have  3a:4-5a;— a;=8a;— x=:(8— 1)  a;=:7a;;  5y— 
y— 3y  =  (5  — 1— 3)y=:(5— 4)yr=y;  (^+26^=^(1  +2)d=z3d\ 
5xy—2xy=('^—2)xy—3xy\  zx\di3ax-\-lax={3-\-l)ax  —  lOax', 
besides  which  there  are  three  quantities  4-4a,  4-yS  +4a;2; 
which  are  unlike,  and  do  not  coalesce  with  any  of  the  others  ; 
the  sum  required  therefore  is, 

7a:+y4-3J+3a;y4-lT)<za:+4a4-y2+4a;2. 
In  Ex.  3.  Beginning  with  40;^,  we  have, 
4a;3  +  5a?3-2a;3=(4-{-5-2)a3  =  (9-2)a;3  =  7a;3; 
—  3ay4-3a:y  — 5a:y=:(3  — 5  — 3)a:y=(3  — 8)cry=:— 5a;y  ; 
+  3y+5y~7y=(3  +  5)y-7y  =  (8-7)v=+y; 

-3  +  30-8=30-(8  +  3)  =  30-ll=::4-19  ; 
2a;2~3a;"'^  — 3a;24-6a;2=.8a;2— (3  +  3)a:2  =  (8  — 6)a:2=:  +2a;2  ; 
5y2— 3y2~2y2  =  5y2— (3  +  2)y2:z.(5-5)y2=0Xy2=0;    -\-2x 
—2x. 

When  quantities  with  literal  coefficients  are  to  be  added 
together  ;  such  as  mx^  ny,  px"^^  qr/,  &c.  (where  m,  n,p,  q,  &c., 
may  be  considered  as  the  coefficients  of  x,  y,  a;2,  y2,  &c.)  it  may 
be  done  by  placing  the  coefficients  of  like  quantities  one  after 
another  (with  their  proper  signs),  under  a  vinculum,  or  in  a 
parentheses,  and  then  annexing  the  common  quantity  to  the 
sum  or  difference. 

Ex.  4. 
ax-\-hy-\-   h 
hx-\-dy-\-2h 

(«+%+(Z>+%'+3& 

Ex.*  5. 

ax^-\-hx'^-\-cx 

ex^ — dx'^—fx  • 


(a^e)x^+{h-d)x'^-\-{c-^f)x 


In  Ex.  4.  The  sum  of  ax  and  hx,  or  ax-^-hx^  is  expressed  by 
{a-\-h)x  ;  the  sum  of  -{-hy  and  -\-dyf  or  -\-by-\-dy,  is  =  + 
{h^d)y,  3* 


18  SUBTRACTION. 

In  Ex.  5.  The  sum  of  arc^  and  ea:^,  or  ax'^-\-ex^is  ={a-\-e)x^; 
the  sum  of  -\-bx^  and  —dx^,  ov-^-bx"^  —dx^,  is  =^{b—d)x''^ ;  and 
the  sum  of  -{-ex  and  —fx,  or  -\-ex-^fx,  is  =z-\-(c—f)x.  Any 
multinomial  may  be  expressed  in  like  manner,  thus  ;  the  multi- 
nomial mx^-{-nx^—px''^—'qx'^  may  be  expressed  by  {m-^-n—p—q) 
a?2-;  and  the  m-ixed  multinomial ^x^y+^y^ — rxy-\-m7/^ — nxy,  by 
(p—r — n)xy-\-{q-\-m)y'^  \  &c. 

Ex.  6.  Add  2a2+y2_j_9^  Ixy—^ab—x'^,  4xy—y—9,  and 
aj2y — xy-^-'Sx"^  together. 

Ans.  4x'^i-y^-\-lOxy—3ab—y-\-x^y. 

Ex.  7.  Add  together  72a^,  2ibc,  70xy,  -18^2,  and— 12ic. 

Ans.  54a--^VZbc-\-70xy. 

Ex.  8.  What  is  the  sum  of  43xy,  7a;2,  —I2ay,  ~4ab,  —3x^, 
End  —  4ay  ?  Ans.  43xy-\-4x'^  —  l6ay—4ab. 

Ex.  9.  What  is  the  sum  of  7xy,  —16bc,  —I2xy  ;  186c,  and 
lixy  ?  Ans.  2bc, 

Ex.  10.  Add  together  5ax,~60bc,  7ax,-^4xy,  —6ax,  and 
—  326c.  Ans.  6ax  —  72bc—4xy. 

Ex.  11.  Add  8a2a;2_3oa;,  Tax— 5a:y,  9a;y— 5aa;,  and  a:y-f 
Zd^x^  together.  Ans.  l0a^x^—ax-\-5xy. 

Ex.   12.  Add  2a;2— 3y2-f-6,  9a;y  — 3aa:— a:2,  4y2_y— 6,  and 
w^y—3xy-{-3xr^  together.     Ans.  4x^-{-y'^-\-6xy  —  3ax—y-{-xy. 
II  1 

Ex.  13.  A.M2x^—4x^-\-x^,  5x'^y—ab+x^,  4x^  —  x^,  and 

2ac3— 3+2x2  together. 

J.       1 
Ans.  4x^—x^-\-5x'^-^5x'^y—ab — x^ — 3. 

Ex.  14.  Requiredthesumof4a;2+7(a4-^P,4y2— 5(.a  +  6)2, 

and  a^—4x^—3y^—(a-]-b)\  Ans.  a3+y2+(a+6)2 

Ex.  15.  Required  the  sum  of  ax^—bx^-\-cx'^,  bcx^—acx^^ 

c^x,  and  ax^-\-c—bx. 

Ans.  aa:* — (b-\-ac)x^-{-{c-}-bc^a)x^ — {c'^+b)x-{-c. 

Ex.  16.  Required  the  sum  of  5a-f  36— 4c,  2a— 56+6c-|- 

2d,  a—4b—2c-\-3e,  and  7a+46— 3c— 6c. 

Ans.  15a— 26— 3c+2(i— 3c. 

§  II.  Subtraction  of.  Algebraic  Quantities, 

62.  Subtraction  in  Algebra,  is  finding  the  difference  be- 
xween  two  algebraic  quantities,  and  connecting  those  quanti- 
ses together  with  their  proper  signs  ;  the  practical  rule  for 
performing  the  operation  is  deduced  from  the  following  propo' 
sition. 

63.   To  subtract  one  quantity  from  another,  is  the  same  thing  as 
to  add  it  with  a  contrary  sign.     Or,  that  to  subtract  a  post' 


SUBTRACTION.  19 

five  quantity/,  is  the  same  as  to  add  a  negative  ;  and  to  sub' 
tract  a  negative,  is  the  same  as  to  add  a  positive. 

Thus,  if  3a  is  to  be  siibtracted  from  8a,  the  result  will  be 
8a  — 3a,  whi'ch  is  5a  ;  and  if  ^  —  c  is  to  be  subtracted  from  a, 
the  result  will  be  a—{h  —  c),  which  is  equal  to  a~b-\-c  :  For 
since,  in  this  case,  it  is  the  difference  between  b  and  c  that  is 
to  be  taken  from  a,  it  is  plain,  from  the  quantity  b — c,  which 
is  to  be  subtracted,  being  less  than  b  by  c,  that  if  b  be  onfj' 
taken  away,  too  much  will  have  been  deducted  by  the  quan- 
tity c  ;  and  therefore  c  must  be  added  to  the  result  to  make  it 
correct. 

This  will  appear  more  evident  from  the  following  conside- 
ration ;  Thus,  if  it  were  required  to  substract  6  from  9,  the  dif- 
ference is  properly  9  —  6,  which  is  3  ;  and  if  6—2  were  sub- 
tracted from  9,  it  is  plain  that  the  remainder  would  be  greater 
by  2,  than  if  6  only  were  subtracted  ;  that  is,  9 — (6 — 2)  =  9 
—6  +  2  =  3  +  2  =  5,  or  9-6  +  2  =  9—4  =  5. 

Also,  if  in  the  above  demonstration,  b—c  were  supposed  ne- 
gative, or  b  —  c=—d  ;  then,  because  c  is  greater  than  b  by  d, 
reciprocally  c—b  =  d,  so  that  to  subtract  —d  from  a,  it  is  ne- 
cessary to  write  a+rf. 

64.  The  preceding  proposition  demonstrated  after  the  man- 
ner of  Gamier, 

Tl>us,  if  fi— c  is  to  be  subtracted  from  the  quantity  a  ;  we 
will  determine  the  remainder  in  quantity  and  sign,  according 
to  the  condition  which  every  remainder  must  fulfil  ;  that  is,  if 
one  quantity  be  subtracted  from  another,  the  remainder  added 
to  the  quantity  that  is  subtracted,  the  sum  will  be  the  other 
quantity.  Therefore,  the  result  will  be  a — b-\-c,  because  a—b 
+  C+6  — c=ra. 

This  method  of  reasoning  applies  with  equal  facility  to  com- 
pound quantities  :  in  order  to  give  an  example  ; 
suppose  that  from  6a— 3i+4c, 
we  are  to  subtract,  5a  — 5Z>+6c  ; 
designating  the  remainder  by  R,  we  have  the  equality, 

R^-Sa— 5i+6c=6a— 3*  +  4c: 
which  will  not  be  altered  (Art.  49.)  by  subtracting  5a,  adding 
5b,  and  subtracting  6c,  from  each  member  of  the   equality ; 
therefore  the  result  will  be, 

R=6a— 35+4c— 5a+55— 6c,  ^ 
or,  by  making  the  proper  reductions,  ^ 

R=a+25— 2c. 

65.  Another  demonstration  of  the  same  proposition  in  La 
place's  manner. 


20  SUBTRACTION. 

Thus  we  can  write, 

a=za-[-h—h  ....  (1), 
a—c=a—c-{-b  —  b  ....  (2) ; 
80  that  if  from  a  we  are  to  subtract  -f-^  or  — ^,  or,  which  ii 
the  same,  if  in  a  we  suppress  -{-b,  or  —b,  the  remainder,  from 
transformation  (1),  must  be  a—b  in  the  first  case,  and  a-{-b  in 
the  second.     Also,  if  from  a — c  we  take  away  +^  or  — b,  the 
remainder,  from  (2),  will  be  a — c  —  b,  or  a — c-{-b. 
•  66.  Hence,  we  have  the  following  general  rule  for  the  sub- 
traction of  algebraic  quantities. 

RULE. 

Change  the  signs  of  all  the  quantities  to  be  subtracted  into 
the  contrary  signs,  or  conceive  them  to  be  so  changed,  and 
then  add,  or  connect  them  together,  as  in  the  several  cases  of 
addition. 

Example  1.  From  ISab  subtract  I4ab. 

Here,  changing  the  sign  of  I4ab,  it  becomes  — 14aJ,  which 
being  connected  to  I8ab  with  its  proper  sign,  we  have  I8ab 
—  I4ab=z{l8  —  l4)ab  —  4ab.     Ans. 

Ex.  2.  From  lox^  subtract  — lOoj^. 

Changing  the  sign  of  —  IOj:^^  it  becomes  ■i-lOx'^,  which 
being  connected  to  ISx^  with  its  proper  sign,  we  have  ISx^-j- 
10x^=z2dx'^.     Ans. 

Ex.  3.  From  24ab+7cd  subtract  ISab-^-lcd. 

Changing  the  signs  of  l8ab-{-7cd,  we  have  —I8ab—7cd 
therefore,  24ab-\-7cd—l8ab—7cd=6ab.     Ans. 
Or,  24ab-{-7cd 

—  I8ab-7cd 


6ab         Ans. 


Ex.  4.  Subtract7a—55+3aa' from  12a+ 10^4- 13aa:--3aJ, 

12a+l0b+l3ax  —  3ab\ 

Changing  the  signs  of  J  > 

all  the  terms  of  7a — 5b  >  — 7a-\-5b — Sax        j 

•f  3aa: ;  it  becomes,        3 

.-.  by  addition,     5a-{-l5b-i-l0ax—3ab. 

Ex.  5.  From  Sab— 7 ax -j- 7 ab-^ Sax,  take  4ai— 3aa;— 4ay, 

Sab  ^7  ax 
^  7ab-{-Sax 

Changinliie  si^s  of  all  l_^^j,^Sax+4xy 
the  terms  of  4ab—Sax—4xyf  >  ^ 

.♦.by  addition,  6a5—aa?4-4a;y.     Ans. 


SUBTRACTION. 


21 


Ex^  6. 
From  36a— 125+7C 
Take  14a—  4+7c— 8 

Rem.  22a—  8b+8   Ans.  * 

In  the  above  example,  one  row  is  set  under  the  other,  that 
is,  the  quantities  to  be  subtracted  in  the  lower  line  ;  then, 
beginning  with  14a,  and  conceiving  its  sign  to  be  changed,  it 
becomes  — 14a,  which  being  added  to  36a,  we  have  36a  — 
14a=:22a  ;  also,  — 46,  with  its  sign  changed,  added  to  — 126, 
will  give  Ab  —  I2b={4  —  I2)b=—Sb  ;  in  like  manner,  7c— 7c 
=0,  and  — 8,  with  its  sign  changed,  =  +  8.  The  following 
examples  are  performed  in  the  same  manner  as  the  last. 
Ex.  7.  Ex.  8. 

From  3a:— 4a4-   6  a+   b 

Take  2x-{-3a-7b  a—  b 


Rem.    x—7a-{-8b 


'+2b 


Ex.  9. 
From  3ab—4:cx-\-  y 
Take  \ax^2x^  —  3y^ 


Rem.  3ab—\ax-\-y—^cx—2x?'Ar3'ip- 


Ex.  11. 
From  bx"^ — 4a;y-{-5 
Take  4x2— 4a:y4-9 


Rem. 


—4 


Ex.  10. 

7x^-\-?,x^—x 
6a;3_2a:2-i-8a; 

x'^-\-bx'^ — 9aj 

Ex.  12. 

7a:2-8 
9a;2+5a6-3a;3 

3x3— 2a;2  — 5a6  _8 


Ex.  13. 
From  ax^ —bx'^  ■\-  x 
Take  px"^ — cx"^  -\-  ex 


(a— p)x3  — (6— c)a;2+(l  — e)a? 


Ex.  14. 
From  bx^-\-qx'^^rx-^py'^ 
Take  ax^ — cx"^ + mx — sy"^ 


(b—a)x^-\-(q-\-c)x'^—{r-\-m)x-\-(p-\-s)y'^ 
67.  As  quantities  in  a  parentheses,  or  under  a  vinculum,  are 


22  MULTIPLICATION. 

considered  as  one  quantity  with  respect  to  other  symbols 
(Art.  10,)  the  sign  prefixed  to  quantities  in  a  parentheses  af- 
fects them  all ;  when  this  sign  is  negative,  the  signs  of  all 
those  quantities  must  be  changed  in  putting  them  into  the  pa- 
renthaees. 

Thus,  in  (Ex.  13),  when  —cx^  is  subtracted  from  —hoc^,  the 
result  is  — bx^-\-cx^,  or  — [h  —  c)x'^  :  because  the  sign  — pre- 
fixed to  {b—c)  changes  the  signs  of  b  and  c  ;  or  it  may  be  writ- 
ten -{-(c—b)x'^. 

Again,  in  (Ex.  14),  when  +ma;  is  subtracted  from  — rx, 
the  result  is  —  rx — mx  ;  and,  as  this  means  that  the  sum  of  rx 
and  mx  is  to  be  subtracted,  that  negative  sum  is  to  be  express- 
ed by  — (rx-\-mx)z=z  —  {r-{-m)x.  For  the  same  reason,  the 
multinomial  quantity — my'^-\-n^y^  — aby^  — ry^-j-Gy^,  when  put 
into  a  parentheses,  with  a  negative  sign  prefixed,  becomes 
—  {m — n'^-\-ab-{-r — 6)y". 
Ex.  15.  From  a  —  b,  subtract  a-\-b.  Ans.  —2b. 

Ex.  16.  From  7a^y— 5y+3a;,  subtract  3a:y+3y+3a;. 

Ans.  Axy—Sy 
Ex.  17.  What  is  the  difference  between  7aa;^4-5a:y—12ay 
-j-5^c,  and  4ax^-{-5xy — 8ay — 4cd. 

Ans.  Sax"^ — 4ay-\-5bc-{-Acd. 
Ex.   18.  From  8x'^  —  3ax-^5,  take  5x^-{-2ax-\-5. 

Ans.  3x^  —  5ax. 
Ex.   19.  From  a-f-i+c,  take  —a— b—c. 

Ans.  2a-\-2b-\-2c. 
Ex.  20.  From  the  sum  of  3a;3— 4aa?+3y2^  Ay'^-\-^ax—x^^ 
yi- — ax-\-bx'^,  and  3aa: — 2x^ — y"^  \    take  the   sum  of  Sy^ — x^ 
4-a;^,  ax—x'^-\-A.x^,  Zx"^ — ax  —  ^y"^,  and  ly'^ — ax-\-l . 

Ans.  4a;^-l-4aa;— 2y2— 5a;2— 7. 
Ex.  21.  From  the  sum  of  x'^y'^—x^y-^xy'^,  Oxy"^  — 15  — 
Sx'^y^,  and  70-{'2x'^y'^—3x^y,  subtract  the  sum  of  5x^y^—20 
~\-xy^,  Sx'^y — x'^y'^-\-ax,  and  Sxy"^  —  Ax'^y'^  —  Q  +  a-rr^. 

Ans.  2xy'^  —  7x'^y  —  ax  —  a'^x'^-\rS4. 
Ex.  22.  From  a^x'^y'^  —  m'^x^  -f  3cx— 4x^  —  9  :  take  a^x^y^ 
—n^x^  +  c'^x-\-bx^-i-3. 

Ans.  {a^^a^)x^y^-{m^-n'^)x^+{3c-^c^)x-{4  +  h.) 

a;2-12. 
^  III.  Multiplication  of  Algebraic  Quantities. 

In  the  multiplication  of  algebraic  quantities,  the  following 
propositions  are  necessary  to  be  observed. 

68.  When  several  quantities  are  multiplied  continually  together, 
the  product  will  be  the  same,  in  whatever  order  they  are  muU 
tiplied. 


MULTIPLICA^TION.  23 

Thus,  axh  —  bXa=ah. 

For  it  is  evident,  from  the  nature  of  multiplication,  that  the 
product  contains  either  of  the  factors  as  many  times  as  the 
other  contains  an  unit.  Therefore,  the  product  ab  contains 
a  as  many  times  as  h  contains  an  unit,  that  is,  h  times. 

And  the  same  quantity  ah,  contains  b  as  many  times  as  a 
contains  an  unit,  that  is,  a  times.  Consequently,  axb  =  ba=: 
ab  ;  so  that,  for  instance,  if  the  numeral  value  of  a  be  12,  and 
of  6,  8,  the  product  ab  will  be  12x8,  or  8x12,  which,  in 
either  case,  is  96. 

In  like  manner  it  will  appear  that  abc=cah  =  bca,  &c. 

69.  If  any  number  of  quantities  be  multiplied  continually  tO' 
getker,  and  any  other  number  of  quantities  be  also  multiplied 
continually  together,  and  then  those  two  products  be  multiplied 
together  ;  the  whole  product  thence  arising  will  be  equal  to 
that  arising  from  the  continual  multiplication  of  all  the  single 
quantities. 

Thus,  ah x cd=a  xbXcX  d=abcd. 
FoT  ab  —  axb,  B,nd  cd—cxd;  if  x  be  put =c(/,  then  ab  X 
cd=abXoc=zaxbXoc ;  but  x  is  =cd=cXd,  .-.  abxx=abxc 
X  d  =  aX  h  X  cd—abcd. 

70.  If  two  quantities  be  multiplied  together,  the  product  will  be 
expressed  by  the  product  of  their  numeral  coe^cients  with  the 
several  letters  subjoined. 

Thus,  7ax5b=::35ab. 

For  7a  is=:7xa,  and  5b=5xb,  .'.7ax5b=7  XaX5xb 
z=z7  X  5  X  «  X  b  =  35  X  ab—35ab. 

71.  The  powers  of  the  same  quantity  are  multiplied  together  by 

addi?ig  the  indices. 

Thus,  to  multiply  a^  by  a^,  it  is  necessary  to  write  the  let- 
ter a  only  once,  and  to  give  it  for  an  exponent  the  sum  2-f  3, 
the  exponents  of  the  factors;  that  is,  a'^Xo^  — a^+^  —  ^s  • 
because  a'^  —  axa,  and  a^=a X «  X a  ;  therefore  a- X a^  — « X a 
XaXaXa  =  a^.  In  general,  the  product  of  a"*  by  a"  ,  m  and  n 
being  always  entire  positive  numbers,  is  a'"+'»  .  In  fact,  a^  is 
the  abbreviation  of  aXaXa,  &c.,  continued  to  m  factors,  and 
a"  is  aXaX a,  <fcc.,  continued  to  n  factors  ;  therefore  u^  X a" 
=  aXaXaXaXa,  &c.,  continued  to  m-^n  factors;  which 
(Art.  12)  is«'»+''  . 

Reciprocally  o"»  +»  can  be  replaced  by  a*"  X  a"  .  The  quan- 
tity a*"  is  sometimes  called  an  exponential. 


24  MULTIPLIpATION 

72.  If  two  quantities  having  like  signs  are  multiplied  together ^ 
the  sign  of  the  product  will  be  +  ;  if  their  signs  are  unlike^ 
the  sign  of  the  product  will  be — . 

1.  A  positive  quantity  being  multiplied  by  a  positive  one, 
the  product  is  positive  ;  thus  +aX  4-^  =  + o^,  because  +a 
is  to  be  added  to  itself  as  often  as  there  are  units  in  6,  and 
consequently  the  product  will  be  -\-ab. 

2.  A  negative  quantity  being  multiplied  by  a  positive  one, 
the  product  is  negative  ;  thus,  —aX  +b=—ab ;  because— a 
is  to  be  added  to  itself  as  often  as  there  are  units  in  b,  and 
therefore  the  product  is  —ab.  Or,  since  adding  a  negative 
quantity  is  equivalent  to  subtracting  a  positive  one,  the  more 
of  such  quantities  that  are  added,  the  greater  will  the  whole 
diminution  be,  and  the  sum  of  the  whole,  or  the  product,  must 
be  negative. 

3.  A  positive  quantity  being  multiplied  by  a  negative  one, 
the  product  is  negative;  thus,  -Jf-aX —bz=:—ab ;  because 
-{-a  is  to  be  subtracted  as  often  as  there  are  units  in  b,  and 
consequently  the  product  is  — ab. 

4.  A  negative  quantity  being  multiplied  by  a  negative  one, 
the  product  is  positive  ;  thus,  —ax—b  =  -{-ab.  For,  aX—b 
=:—ab,  that  is,  when  the  positive  quantity  a  is  multiplied  by 
the  negative  quantity  b,  the  product  indicates  that  a  must  be 
subtracted  as  often  as  there  are  units  in  b  ;  but  when  a  is  ne- 
gative, its  subtra,ction  is  equivalent  to  the  addition  of  an  equal 
positive  quantity ;  therefore,  in  this  case,  an  equal  positive 
quantity  must  be  added  as  often  as  there  are  units  in  b. 

73.  If  all  the  terms  of  a  compound  quantity  be  multiplied  sepa- 
rately by  a  simple  one^  the  sum  of  all  the  products  taken  to- 
gether, will  be  equal  to  the  product  of  the  whole  compound  quan- 
tity  by  the  simple  one. 

For,  in  the  first  place,  \(  a-\-b  be  multiplied  by  c,  the  pro- 
duct will  be  ca-\-bc  :  Since  a-\-b  is  to  be  repeated  as  many 
times  as  there  are  units  in  b  j  the  product  of  a  by  c,  that  is, 
ca,  is  too  little  by  the  product  of  b  by  c,  that  is,  cb  ;  it  is  ne- 
cessary then  to  augment  ca  by  cb,  which  will  give  for  the  pro- 
duct sought  ca-\-cb,  where  the  term  -\-cb  arises  from  multiply- 
ing +  ^  by  c.  It  would  be  found  by  reasoning  in  like  manner, 
that  the  product  of  c  by  a-[-b  must  be  ca-\-cb,  where  -\-cb  is 
ex  -f-6.  If,  in  the  second  place,  a— i  be  multiplied  (where  a 
is  greater  than  b)  by  c,  the  product  will  be  ca  —  cb.  Since 
a — b  is  to  be  repeated  as  many  times  as  there  are  units  in  c ; 
the  product  of  a  by  c  will  give  too  great  a  result  by  the  pro- 


MULTIPLICATION.  25 

duct  ch  ;  it  is  necessary  then  to  diminish  the  product  ca  by  ch, 
so  that  the  true  product  is  ca—cb. 

Let,  for  example,  7—2  be  multiplied  by  4  ;  the  product  will 
b^8  — 8,  or  20  ; 

For,  7x4,  or  28,  is  too  great  by  2  X  4,  or  by  8  ;  therefore, 
the -true  product  will  be  the  first  diminished  fey  the  second,  or 
28— 8,  that  is  20.  In  fact,  7— 2,  or  5x4=20.  The  term 
— cb  of  the  product,  is  the  product  of  — b  by  c. 

It  would  be  found,  by  reasoning  in  like  manner,  that  the 
product  of  c  by  a-mb,  must  be  ac—bc,  the  same  as  in  the  pre- 
ceding, and  in  which  the  term  — be  is  the  product  of  c  by  — b. 

If,  in  the  third  place,  a-\-b-\'d  be  multiplied  by  c,  the  pro- 
duct will  be  ca-\-cb-{-cd. 

For,  let  a-\-b  be  designated  by  e;  then,  e+d  multiplied  by 
c  is  equal  to  ce+cc?;  t>ut  ce  is  equal  to  cX{a-\-b)=ca-\-cb, 
because  e  is  equal  to  a-\-b  ;  therefore  {a~\-b-\-d)xc  =  ca-{-cb 
4-cJ.  Also,  if  {a-\-b) — d  be  multiplied  by  c,  the  product  will 
be  ca-\-ch—cd;  for  let  (a-f-^)  =  e,  then  (e — d)Xc=ce — cd=z 
c[a-\-b) — cd=ca-\-cb — cd. 

Finally,  it  may  be  demonstrated  in  like  manner,  that  if  any 
polynomial,  a-\-b  —  d-\-e—f,  &c.,  be  multiplied  by  c,  the  pro- 
duct will  be  ca-\-cb — cd-\-ce — cf,  &c.  Also,  if  a  quantity  e 
be  multiplied  by  any  polynomial  a-{-b — d-\-ej  &c.,  the  pro- 
duct will  be  ac-\-bc--dc-\-ec,  &c. 

75.  If  a  compound  quantity  be  multiplied  by  a  compound  quan^ 

tity,  the  product  will  be  equal  to  every  term  of  one  factor,  muU 

tiplied  by  every  term  of  the  other  factor,  and  the  products 

added  together. 

Let,  in  the  first  place,  a-\'b  be  multiplied  by  c-\-d'.  a-\-h 
taken  c  times  is  ca-\-cb,  as  we  have  already  proved  ;  but  this 
product  is  too  little  by  the  binomial  a-\-b  repeated  d  times,  it 
is  necessary  then  to  add  to  it  da-\-db,  and  we  will  have  ca-{-cb 
•\-da-\-db  for  the  product  sought;  in  which  the  term  -{-db 
arises  from  the  multiplication  oi  -\-bhy  -\-d. 

Suppose,  in  the  second  place,  that  a^b  is  multiplied  by 
c — d,  the  product  will  be  ca-\-cb—da — db. 

Because  the  product  of  a-{-b  by  c,  that  is,  ca-\-cb,  is  too 
great  by  that  of  a-\-b  hy  d,  which  is  da-\-db  ;  we  will  have 
therefore  the  true  product  equal  to  ca-\-cb  —  da-mdb,  where  the 
term  — db  is  the  product  of  -\-bhy  — d  ;  in  multiplying  c~d 
by  a-\-b,  we  will  find  that  — bd  is  the  product  of  —d  by  -\-b. 

Let,  in  the  third  place,  a  —  b  he  multiplied  by  c — d  ;  the 
product  will  be  ca'—cb—da+db. 

For,  the  product  of  c— &  by  c,  that  is,  ca—cby  is  too  little  by 
4 


26 


MULTIPLICATION. 


that  of  a—b  by  d,  which  is  da—db  ;  because  the  multiplier  c 
is  too  great  hy  d  \  it  is  necessary  then  to  subtract  the  second 
product  from  the  first,  and  the  difference  will  be  (66)  ca  —  cb 
~da-\-db.  % 

Here  the  term  -\-bd  results  from  — 6  by  — d. 

Finally,  if  a^b-\-e  be  multiplied  by  c-\-d  the  product  will 
be  ca~\-cb-^ce-\-ad-{-bd-\-de. 

For,  in  designating  a-\-b  by  h  ;  then,  {h-^e)x{c-\-d)=zhc-\- 
ec-{  dh-\-ed,  which  is  equal  to  hx{C'i-d)-\-cc-\-ed=(a-^b)X 
[c'i-d)-\-ec-\~ed=ca-{-cb-jrce-{-ad'\-bd-\-del^ 

The  same  mode  of  reasoning  may  be  extended  to  compound 
quantities  composed  of  any  number  of  terms  whatever. 

76.  Cor.  Hence,  in  general,  if  any  two  terms  which  are 
multiplied  have  different  signs,  their  product  must  be  preceded 
by  the  sign  — ,  and  if  they  have  the  same  sign,  the  product 
is  affected  with  the  sign  -j-  ;  agreeably  to  what  has  been  de- 
monstrated (Art.  72.)  where  simple  quantities,  or  isolated  fac- 
tors, such  as,  4- a,  -j-^,  — «,  — b,  were  only  considered. 

From  the  division  of  algebraic  quantities  into  simple  and 
compound,  there  arises  three  cases  of  Multiplication :  the 
practical  rules  for  performing  the  operation  are  easily  deduced 
from  the  preceding  propositions. 

CASE  1. 

When  the  factors  are  both  simple  quantities. 


RULE. 


77.  Multiply  the  coefficients  together,  to  the  product  suo- 

join  the  letters  belonging  to  both   the  factors,  and  the  result, 

with  the  proper  sign  prefixed,  will  be  the  product  required. 

Ex.  1.  Ex.  2.  Ex.  3.  Ex.  4. 

Multiply         Sab  5x  —  6y  —Aa^ 

By         4c  —  3a  +3x  —6x^ 


Product 

I2abc 

Ex.«. 

2ax 
—Sax 

—  15a.T 

—  18a;y 

Ex.7. 

xY 
-Ixy 

—7xY 

'\-24a^x^ 

Mul. 
By 

Ex.  6. 

—Sa^c 
+  5ac2 

Ex.  8. 

'-5a^^c 

-4a^^-x 

Pro.       - 

-I6a^x^ 

— ISaV 

+20a*b^cx 

MULTIPLICATION,  27 

Ex.  9.  Required  the  product  of  Aahc  and  2a^c. 

Ans.  i2c^bc\ 
Ex.  10.  Required  the  product  of  —laxy  and  —2acx. 

Ans.  -\- ^'^o.^cx^y 
Ex.  11.  Required  the  product  oUx'^y^  and  —  Sy^a;-*. 

Ans.  —  21a;^y''. 
Ex.  12.  Required  the  product  of  a^  and  —a^.  Ans.  — a^. 
Ex.  13.  Required  the  product  of  aa?^  and  hx'^z. 

Ans.  ahx^z^. 
Ex.  14.  Required  the  product  of  — xyz  and  ahc. 

Ans.  — ahcxyz. 
Ex.  15.  Required  the  product  of  —^h'^cd?'  and  —2a^hc^d. 

Ans.  8a3^*3c3<Z3 
Ex.  16.   Required  the  product  of  — Sa^  and  Aa. 

Ans.  —12a* 
Ex.  1 7.  Required  the  product  of  a^b^c  by  a^c'^d. 

Ans.  a^b*c^d 

CASE  II. 

WAcn  one  factor  is  Compound  and  the  other  Simple, 

RULE. 

78.  Multiply  each  term  of  the  compound  factor  by  the  sim- 
ple factor,  as  in  the  last  case  ;  then  these  products  placed 
one  after  another  with  their  proper  signs,  will  be  the  product 
required. 

Ex.  1. 
Multiply  4xy—3ax-\-2y 
by  4ax 


Product  l6ax^y—l2a^x^-\-  Saxy 


Ex.  2. 
Mul.     4a;3_3a;2_8 
by  —^ax 


Pro.  —Sax^-\-Qax^-\-\Qax 

Ex.  3. 
Mul.  8a3-752+3a_l 
hy2b 

Pro.  \Qa^b^\4a'^b-\-Qab—2b 


28  MULTIPLICATION. 

Ex.  4. 
Mul.     3a;2y02_a.y2^_2a2y 

by  —x^yz 


Pro.  —  3a;y^3+^y-s^  +  2a2a;2y2^ 

Ex.  5.  Multiply  Qa^x^—^h  +  c  by  2ac. 

Ans.  16a3ca;2— 6aic  +  2ac2. 
Ex.  6.  Multiply  —  Sa:^— 4a2-}-5  by  —  4flra:. 

Ans.  l2ax^-\-lQa^x~20ax. 
Ex.  7.  Multiply  a^-j-aar+a?^  by  ax. 

Ans.  a3a;_|-a,2a;2_|_^a,3^ 
Ex.  8.  Multiply  x'^  —  xy-^y'^hy  —x^y. 

Ans.  — x*'y-^x^y'^-\-x^y^. 
Ex.  9.  Multiply  3a2_2a^>+3^>2  by  a2^,2. 

Ans.  3a452— 2a3&3+3a2K 
Ex.  10.  Multiply  a2a;2_Qa;_|_9by5.  K\i%.^d^x'^  —  bax'\-\^. 
Ex.  11.  Multiply  2c(f—3a2>— 3  by  4ac. 

Ans.  8ac2c;— 12a2ic— 12ac. 
Ex.  12.  Multiply  7a7-3  +  3c&— 5y2  ^y  —xy. 

Ai\s.  —  7x'^yz—3abxy-{-5xy^. 
Ex.  13.  Multiply  a+Z>  —  c—cZ  by  aSc</. 

Ans.  a?bcd-\-ah'^cd—abc'^d—abcd^ 


CASE  III. 
When  both  factors  are  compound  quantities, 

RULE. 

79.  Multiply  every  term  of  the  multiplicand  by  each  term  of 

the  multiplier  successively,  as  in  the  kst  case  ;  then,  add  or 
connect  all  the  partial  products  together,  and  the  sum  will  be 
the  product  required. 

Note.  It  is  necessary  to  observe  that  like  quantities  are  ge- 
nerally placed  under  each  other,  in  order  to  facilitate  their  addi- 
tion. And  if  several  compound  quantities  are  to  be  multiplied 
continually  together  ;  thus, 

[a+b)  X  (a-b)  X  {o?+ab+b'')  X  (02-0^4.^2). 
Multiply  the  first  factor  by  the  second,  and  then  that  product 
by  the  third,  and  so  on  to  the  last  factor ;  but  it  is  sometimes 
more  concise  not  to  observe  the  order  in  which  the  compound 
quantities,  or  factors,  are  placed,  as  can  be  readily  seen  from 
the  following  examples. 


MULTIPLICATION.  29 


EXAMPLE  1. 

Multiplicand      2a'^—3ba^—5b^a^ 
Multiplier  a^—2ba^-\-3b^a 


1st  partial  pro.  2a''~3ba^—5b^a^ 

second  ''4ba^-{-6h^a^-\-l0b^a^ 

third  -f  66V-9i3a*-15Ma3 

Total  prod.     =2a'^-7ba^-{-7b^a^-{'  b^a*—l5b^a^ 


Ex.2. 

Multiply  a+b 

by  a — b 


Ist  partial  prod,     a^-^ab 
second  ^ab—b^ 


Total  product         a^     *  —b^ 


Ex.  3. 

Multiply  a^^ab+b^ 
by  a^—b"^ 


Ist  partial  product  a'^-\-a^b-\-a^b^ 

second  —d^b'^—ab^—b* 


Total  prod.  a^-\-a^      *     —ab^  —  b^ 


E!t.  4. 
Multiply  a*+a36— a53— 5* 
by  a^—ab  +b^ 


1st  partial  prod,  a^-^a^b—a^^—d^b* 
second  ^a^b—a'^b^  +  a^b^-^-ab^ 

third  +a^b^+a^^—ab^—b^ 


Total  product     a^        *        *        *         *  — J6 


Ex.  5. 

Multiply  a2^c5_^j2 

by  a^'-ab  \-b'^ 


30  MULTIPLICATION. 

1  St  partial  prod,  ^-\-a^+ a^b"^ 
second  —a^—aW—ab"^ 

third  -\-aW-\-ab^+b^ 


Total  prod.          a*     *    +02^2      *  .j-j* 

Ex,  6. 

Multiply  a*+a2&2_j_j4 
by  a2_^2 

1st  partial  product  a^-{-a'^b'^-{-a%^ 
second                          —a'^b'^—a'^b^—b^ 

Total  product           a^       *         *    —b^ 

Ex.7. 

Multiply  a2_^aj_j_j2 
by  a  —b 

I  St  partial  prod.      a^  +  a'^b  ^ab^ 
second                          —a^b—ab^—b^ 

Total  product          a^      *       ♦    —P 

Ex.  8. 

Multiply  a^—ab+b^ 
by  a  -\-b 

first           a^-^a^-^ab^ 
second        -{-a^b—ab^+b^ 

Product   a^      *        *  -{-b^ 

Mul. 
by 

Ex.  9.                                       Ex.  10. 

a3+i3                                      a  -b 

1st. 
2nd. 

Prod. 

a«      ♦    -56                           p3-2a2i+2fl62-63 

MULTIPLICATION.  31 

Ex.  11. 

Multiply  a^+ab-^b^ 
by  a  -{-b 


first  a^-\-a^-\-ab^ 

second  -{•oF^b-j-ab^-^-P 


Product  a^-\-2a^b+2ab^-^b^ 

Ex.  12. 

Mult.  a^-\-2aH  +  2ab^+P 
by  a^—2a^+2al)^—b^ 

1st.  a^-\-2a^b-{-2a^b'^-\-   a^b^ 

2nd.        —2a^b'^4a^b'—4aH^—2a^b* 

3d.  4-2a462+4a363+4a26*+2a&5 

4th.  —  a?P—2a^b^—2ab^  —  ¥ 

prod.  a6         *         *  .*  *  *     — i« 

When  the  quantities  to  be  multiplied  together  have  literal 
coefficients,  proceed  as  before,  putting  the  sum  or  difference 
of  the  coefficients  of  the  resulting  terms  into  a  parentheses,  or 
under  a  vinculum,  as  in  Addition. 

Ex.  13. 
Mult.  x'^—ax-\'p 

by  a;24-ix-f  3 


1st.    x^—ax^-^-px^ 

2nd.      ■\-bx^ — abx'^-\-bpx 

3d.  +3af2— 3aa;+3;) 


prod,  a;*— (a— 6)a;34-(;>-.a&+3)a;2+(6p— 3a)a?+3;) 

Ex.  14. 
Mult,  ax'^—  bx  -^-c 

by    x^ —  ex  -\-\ 


1st.     ax'^-^  bx^-\-  cx^ 
2nd.         — acx^+bcx^ — c^x 
3d.  +  oa;^  —  bx-{-c 


prod.  ax^--(b  +  ac)x^~\-(c-]-bc+a)x^—(c^-{-b)+c 

Ex.  15.  Required  the  continual  product  of  a+2a?,  a— 2ap, 
and  a2+4a:2. 


32  MULTIPLICATION. 

Multiply  a-\-2x 
by  a—2x 


a^-\-2ax 
—2ax- 

4x^ 

Multiply 
by 

a2_4a:2 
a2+4a;2 

a4-.4a2a;2 
+4a2a;2. 

-16a;4 

Total  product 

a*        * 

-16a;* 

It  may  be  necessary  to  observe,  that  it  is  usual,  in  some 
cases,  to  write  down  the  quantities  that  are  to  be  multiplied 
together,  in  a  parentheses,  or  under  a  vinculum,  without  per- 
forming the  whole  operation  ;  thus,  {a-{-2x)  X  ia—2x)  x  {a^-\- 
Ax"^).  This  method  of  representing  tlie  multiplication  of  com- 
pound quantities  by  barely  indicating  the  operation  that  is  to 
be  performed  on  them,  is  preferable  to  that  of  executing  the 
entire  process  ;  particularly  when  the  product  of  two  or  more 
factors  is  to  be  divided  by  some  other  quar»tity ;  because,  in 
this  case,  any  term  that  is  common  to  both  the  divisor  and 
dividend  may  be  more  readily  suppressed  ;  as  will  be  evident, 
from  various  instances,  in  the  following  part  of  the  work. 
Ex.  16.  Required  the  product  of  a-f-^  +  cby  a  —  b-\-c. 

Ans.  a2_|_2ae  — Z>2-|-c2. 
Ex.  17.  Required  the  product  of  xH-yH-2^  by  a?— y—;^. 

Ans.  x'^  —  \p-~2yz—z^. 
Ex.  18.  Required  the  product  of  I— a;+a;2— a:^  by  l+a;. 

Ans.  1— a:*. 
Ex.  19.  Multiply  a3 4- 3a25-j- 3^524- 53  by  a2 4. 2a6-hR 

Ans.  a5^5a*6+10a352_j_i0a2^3_^5g^4_|_^5, 
Ex.  20.  Multiply  4a;2y  + 3 j:y  —  l  by  2a;2— a;. 

Ans.  8a;*y+2a:3y— 2ar2  — 3a:2y+a;. 
Ex.  21.  Multiply  a:3-}-a;2y4-iry2-{-y3by  a?— y.  Ans.  x^—y^. 
Ex.  22.  Multiply  Zx^—2cP-x^-^'id^  by  2x^  —  ^a^x'^-\-ba?. 

Ans.  6x«  — 13a2x5+6tf%*+21fl3x3— 19«^a;2H-15a6. 
Ex.  23.  Multiply  2a2—3aa;  + 4x2  by  5a2_ 6arc— 2x2. 

Ans.  lOfli— 27a3a:+34a2a;2— I8ax3— 8x*. 

Ex.  24.  Required  the   continual  product  of  a-\-x,  a— x,  c? 

4-2aa;-|-a2,  and  c^ — 2ax-f-a;2.        Ans.  a^ — 3a'*x24-3a2ar*— x®. 

Ex.  25.  Required  the  product  of  «^— aa2-f-Jx— c,  and  o? 

— 2a:+3. 


DIVISION.  33 

Ans.  a;5-(a+2).r*+(&+2a+3)a;3-(c4-2i  +  3a)x2+(2c4.3*) 
x—2c. 

Ex.  26.  Required  the  product  oi'  mx^—nx — r  and  nx — r. 

Ans.  mnx"^ — (n?--{-mr)x'^-\-r^. 
Ex.  27.  Required  the  product  of  px"^ — rx-^-q  and  x"^ — rx 
— q.  Ans.  px^ — {r-\-pr)x^-\-{q-\-r^^pq)x'^—q^. 

Ex.  28.  Multiply3a;2— 2a;y  +  5  by  a:2-f2a:y— 3. 

Ans.  3a;'*+4T3y_4a:2  X  {\^-y'^)+lQxy—\b. 

Ex.  29.  Multiply  a^  +  3a^b  +  3a&2  4.  53  by  a^-^a^h  + 

3ab'^—¥.  Ans.  a^— 30*62  _f-3a254_^6, 

Ex.  30.  Multiply  5a3—4a254-5aZ>2__3^,3  by  4a2_5a^,4.2i^2. 

Ans.  20a5— 41a'*^>+50a362— 4.5a2J3^25ai*— 655. 

Ex.  31.  Required  the  continual  product  of  a-\'X,  a'^-\-2ax 

4-a;2,  and  a'^-\-^a'^x-\-'^ax'^-\-x^,         » 

Ans.  a6-f6a^''x+15a%2_^20a3a;3-f-l5a2a;4-f  6aa;5+a;6. 

Ex.  32.  Required  the  continual  product  of  a — ar,  a^ — 2ax-\- 
r2  and  a^ — 3a2a;4-3aa;2 — x^. 

Ans.  a^—6a^x-{-i5a*x^—20a^x^+\5a*x^—6ax^+x^. 

^  IV.  Division  of  Algebraic  Quantities. 

80.  In  the  Division  o(  a.]gehrB.ic  quantities,  the  same  circum- 
stances are  to  be  taken  into  consideration  as  in  their  multipli- 
cation, and  consequently  the  following  propositions  must  be 
observed. 

81 .  If  the  sign  of  the  divisor  and  dividend  be  like,  the  sign  of  the 
quotient  will  be  -\-  ;  if  unlike,  the  sign  of  the  quotient  will  be — . 

The  reason  of  this  proposition  follows  immediately  from  mul- 
tiplication. 

Thus,    if    +ax+b=z+ab  ;     therefore  ^tf_=-f 5 

-\-aX—b=—ab;         .-.  ^^^=—b 

—aX-\-b=—ab;         /.  =  +  J 

—  a 

^aX-b=+ab;         .-.  ±^=_J 

— a 

82.  If  the  given  quantities  have  coefficients,  the  coefficient  of  the 
quotient  will  be  equal  to  the  coefficient  of  the  dividend  divided  by 
that  of  tJie  'divisor. 

Thus,  4a6-^25,  or  ~=2a. 

<q0 

For,  by  the  nature  of  division,  the  product  of  the  quotient, 
multiplied  by  the  divisor,  is  equal  to  the  dividend ;  but  the  co- 


34  DIVISION. 

efficient  of  a  product  is  equal  to  the  product  of  the  coefficients  ol 
the  factors  (Art.  70).     Therefore,  4ab-^2b  =  -x-j-=2a. 

83.  That  the  letters  of  the  quotient  are  those  of  the  dividend  not 
common  to  the  divisor^  when  all  the  letters  of  the  divisor  are  com- 
mon to  be  dividend  :  for  example,  the  product  abc^  divided  by  ab, 
gives  c  for  the  quotient,  because  the  product  of  ab  by  c  is  abc. 

84.  But  when  the  divisor  comprehends  other  letters,  not  common 
to  the  dividend,  then  the  division  can  only  be  indicated  and  the  quo- 
tient written  in  the  form  of  a  fraction,  of  which  thenumerator  is  the 
product  of  all  the  letters  of  the  dividend,  not  common  to  the  divisor, 
and  the  denominator  all  those  of  the  divisor  not  common  to  the  divi- 

dend  :  thus,  abc  divided  by  amb,  gives  for  the  quotient  — ,  in  ob- 

m 

serving  that  we  suppress  the  common  factor  a^,  in  the  divisor 

and  dividend  without  altering  the  quotient,  and  the  division  is  , 

reduced  to  that  of  — ,  which  admits  of  no  farther  reduction 
m 

without  assigning  numeral  values  to  c  and  m. 

85.  If  all  the  terms  of  a  compound  quantity  be  divided  by  a  simple 

one,  the  sum  of  the  quotients  will  be  equal  to  the  quotient  of  the 

whole  compound  quantity. 

„,        ab  ^    ac   ^  ad      ab+ac-\-ad 

Thus, 1 1 = —b-\-c-\-d. 

a        a        a  a 

For,  (b-\-c-^d)  Xa  —  ab-\-ac-\-ad. 

86.  If  any  power  of  a  quantity  be  divided  by  any  other  power  of 
the  same  quantity,  the  exponent  of  the  quotient  will  be  that  of  the 
dividend,  diminished  by  the  exponent  of  the  divisor. 

Let  us  occupy  ourselves,  in  the  first  place,  with  the  division 

of  two  exponentials  of  the  same  letter ;  for  instance,  — ,  m  and 

n  being  any  positive  whole  numbers,  so  that  we  can  have, 
m^n,  m=n,  m<Cn.  * 

It  may  be  necessary  to  observe  that,  according  to  what  has 
been  demonstrated  (71),  with  regard  to  exponentials  of  the 
same  letter,  the  letter  of  the  quotient  must  also  be  a,  and  if  the 
unknown  exponent  of  a  be  designated  by  x,  then  a*  will  be  the 
quotient,  and  from  the  nature  of  division, 

■from  which  there  necessarily  results  the  following  equality 
between  the  exponents, 


DIVISION.  35 

mz=:n-\-x  ; 
And  as,  subtracting  n  from  each  of  these  equal  quantities,  the 
two  remainders  are  equal  (Art.  49),  we  shall  have 
m  —  n=zx  .  .  .  .  (I). 
Therefore,  in  the  first  case,  where  m  is  >n,  th^exponent  of 
the  quotient  is  m—n  ;  thus, 

a^-^a^=a^-^=za^j  and  a^-^a=a^~^=a^. 
Also,  it  may  be  demonstrated  in  like  manner,  that  (a+a)*-:- 

(a  +  x)^=(a  +  x)^^=(a-\-xy  ;  andA^^^3  =  (2a:+yr-«= 

(2x+yf. 

In  the  second  case,  where  m=n,yve  shall  have, 

From  which  there  results  between  the  *xponents  the  equality, 

m  =  772Hi- a?, 
and  subtracting  m  from  each  of  these  equals  (Art.  49), 

m—m=.x,  or  j:=rO  ....  (2)  ; 
therefore,  the  exponent  of  the  quotient  will  be  equal  to  0,  or 
axz=^a°^  a  result  which  it  is  necessary  to  explain.     For  this 
purpose,  let  us  resume  the  division  of  a"*  by  a'",  which  gives 

unity  for  the  quotient,  or — =1  ;  and  as  two  quotients,  aris- 
ing from  the  same  division,  are  necessarily  equal ;  therefore, 

a«  =  l.  . 

Hence,  as  a  may  be  any  quantity  whatever,  we  may  conclude 
that ;  any  quantity  raised  to  the  power  zero,  must  he  equal  to 
unity,  or  1,  and  that  reciprocally  unity  can  he  translated  into 
a°.  This  conclusion  takes  place  whatever  may  be  the  value 
of  a  ;  which  may  also  be  demonstrated  in  the  following  man- 
ner. 

Thus,  let  a'>—y\  then,  by  squaring  each  member,  0^X0"= 
yXy,  ora°^y2. 

therefore,  (47),  y'^^y^ 
and  (51),-^=^, 

^      y    y 

ory-1; 
but  a°=;y  ;  consequently  a°=:l. 
In  the  third  case,  where  m  is  less  than  n ;  l^^=m+£?,  d 
being  the  excess  of  n  above  m ;  we  shall  alwsjHpave, 

and  equalising  the  exponents,  because  the  preceding  equality 
cannot  have  place,  but  under  this  consideration, 

m=:m-\-d-\-x, 
subtracting  m-\-d  from  both  sides,  the  final  result  will  be 


86  DIVISION. 

x=—d (3); 

then  the  quotient  is  a-^ . 

In  order  to  explain  this,  let  us  resume  the  division  of  a*"  by 
a",  or  by  a^^^zrza^xa^  \  by  suppressing  the  factor  a'",  which 
is  common  to  the  dividend  and  divisor,  according  to  what  has 
been  demonstrated  with  regard  to  the  division  of  letters  (Art. 

84),  we  have  for  the  quotient  -^  :  therefore, 

°^=i (4); 

This  transformation  is  very  useful  in  various  analytical 
operations ;  in  order  to  see  more  clearly  the  meaning  of  it, 
we  may  recollect  that  a^^  is  the  same  as  aXaXa,  &c.,  con- 
tinued to  d  factors  ;  therefore,  according  to  the  acceptation 
and  opposition  of  the  signs,  a-<^  must  represent  aXaXa  &c., 
continued  to  d  factors  in  the  divisor. 

Hence,  according  to  the  results  (1),  (2),  and  (3),  the  pro- 
position is  general,  when  m  and  n  are  any  whole  numbers 

whatever;  thus,  a^4-o^=a^~^=a-2,  or -^ :    because    the  di- 

visor  multiplied  by  the  quotient  is  equal  to  the  dividend,  a^  X 

1  a^ 

a-'^=a^-^=a^=  the  dividend,  and-^Xa^=  ~-r=a^-^=a^= 

(P-  a?' 

,  •  1 

the  dividend,  therefore,  a-2=— -.     In  like  manner  it  may  be 

shown  that,  — =a-3,  —  =  a-*,  &c.     But,  according  to  the 

result  (4),  in  general,  — •=:a-<^,  where  d  may  be   any  whole 

number  whatever  ;  hence  the  method  of  notation  pointed  out, 
(Art.  32),  is  evident. 

87.  If  a  compound  quantity  is  to  be  divided  by  a  compound 
quantity,  it  frequently  occurs  that  the  division  cannot  be  per- 
formed, in  which  case,  the  division  can  be  only  indicated,  in 
representing  the  quotient  by  a  fraction,  in  the  manner  that  has 
been  already  described  (Art.  8). 

88.  But  ifltjjk  of  the  terms  of  the  dividend  can  he  produced  hy 
multiply i^T  the  divisor  hy  any  simple  quantity,  that  simple 
quantity  will  be  the  quotient  of  all  those  terms.  Then  the  re- 
maining terms  of  the  dividend  may  he  divided  in  the  same 
manner,  if  they  can  he  -produced  hy  multiplying  the  divisor 
hy  any  other  simple  quantity ;  and  by  continuing  the  same 


DIVISION.  37 

method,  until  the  whole  dividend  is  exhausted  ;  the  sum  of  all 
those  simple  quantities  will  be  the  quotient  of  the  whole  com- 
pound quantity. 

The  reason  of  this  is,  that  as  the  whole  dividend  is  made  up 
of  all  its  parts,  the  divisor  is  contained  in  the  v^^hole  as  often  as 
it  is  contained  in  all  its  parts.  Thus,  (ah-\-cb-\-ad-\-cd)-r' 
(a-\-c)  is  equal  to  b-{-d: 

Yox  bx{a-\-c)=zab-\-cb\  and  dx(a-\-c)=ad-\-cd]  but  the 
sum  oi  ab-\-cb  and  ad-\-cd  is  equal  to  a6+c^-f  arf-j-cc?,  which 
is  equal  to  the  dividend  ;  therefore  b-^d  is  the  quotient  re- 
quired. 

Also,  (a24-3a5  +  2i2)4.(a+J)  is  equal  to  a+2^». 
For,  it  is  evident  in  the  first  place,  that  the  quotient  will 
include  the  term  a,  since  otherwise  we  should  not  obtain  a^. 
Now,  from  the  multiplication  of  the  divisor  a-\-b  by  o,  arises 
a'^-\-ab  ;  which  quantity  being  subtracted  from  the  dividend, 
leaves  a  remainder  2ab-\-2b'^ ;  and  this  remainder  must  also 
be  divided  by  a-^-b,  where  it  is  evident  that  the  quotient  of 
this  division  must  contain  the  term  2b  :  again,  2b,  multiplied 
by  a-\-  b,  produces  2ab-\'2b^  ;  consequently  a-\-2b  is  the  quo- 
tient required  ;  which,  multiplied  by  the  divisor  a-^b,  ought 
to  produce  the  dividend  a^-f  3a6+262.  See  the  operation  at 
length : 

a-{-b)a^-^3ab-{-2P{a+2h 
a^4-  ob 


2ab+2b^ 
2ab-j-2b^ 


89.  Scholium.  If  the  divisor  be  not  exactly  contained  in 
the  dividend  ;  that  is,  if  by  continuing  the  operation  as  above, 
there  be  a  remainder  which  cannot  be  produced  by  the  mul- 
tiplication of  the  divisor  by  any  simple  quantity  whatever ; 
then  place  this  remainder  over  the  divisor,  in  the  form  of  a 
fraction,  and  annex  it  to  the  part  of  the  quotient  already  de- 
termined ;  the  result  will  be  the  complete  quotient. 

But  in  those  cases  where  the  operation  will  not  terminate 
without  a  remainder  ;  it  is  commonly  most  convenient  to  ex- 
press the  quotient,  as  in  (Art.  87). 

90.  Division  being  the  converse  of  multiplication^  it  also  ad- 
mits of  three  cases. 


38  DIVISION. 

CASE  I. 

When  the  divisor  and  dividend  are  both  simple  quantities, 

RULE. 

91.  Divide,  at  first,  the  coefficient  of  the  dividend  by  that  of 
the  divisor ;  next,  to  the  quotient  annex  those  letters  or  factors 
of  the  dividend  that  are  not  found  in  the  divisor  ;  finally,  pre- 
fix the  proper  sign  to  the  result,  and  it  will  be  the  quotient  re- 
quired. 

Note.  Those  letters  in  the  dividend,  that  are  common  to  it 
with  the  divisor,  are  expunged,  when  they  have  the  same  ex- 
ponent ;  but  when  the  exponents  are  not  the  same,  the  expo- 
nent of  the  divisor  is  subtracted  from  the  exponent  of  the 
dividend,  and  the  remainder  is  the  exponent  of  that  letter  in  the 
quotient. 

Example  1.  Divide  ISax^  by  Sax. 

18ax2      18     a     x^      ^      ,        o    ,      ^ 
3ax         Sax 

Or,    — =:—xa^-^Xx^-^  =  exa'>Xx=ex.     See  (Art 

Sax        3  ^ 

86.) 

Ex.  2.   Divide  —48a^^c^  by  I6ahc, 

In  the  first  place,  484- 16  =  3=  the  coefficient  of  the  quo- 
tient, next,  a'^b^c'^-^abcz=a^~^  Xb'^~^  Xc^~^=:abc  ;  now,  an- 
nexing aic  to  3,  we  have  Sabc,  and,  prefixing  the  sigr» —  ;  be- 
cause the  signs  of  the  dividend  and  divisor  are  unlike  ;  the  re- 
sult is  —Sabc,  which  is  the  quotient  required. 

Or,  the  operation  may  be  performed  thus, 
—48^.262^2  48     a2      52      c2 

— --7 =  —  ■r^X~X-r-X--=—3XaXbxc=  —Sabe. 

Idabc  16     a       0       c 

Ex.  3.  Divide  — 21a;3y%*  by  —Ix^yH'^. 
— 21a;V;g*_      21 
—Ix^y'^z^    ~"^  7 

Ex.  4.  Divide  280^*^  by  — 70252^5. 
28     fl*      W      c^ 
7      a^      b^      c^ 
Xc'-5=:_4xa2X^>3xc2=-4a2^V. 

In  order  that  the  division  could  be  effected  according  to  the 
above  rule  ;  it  is  necessary,  in  the  first  place,  that  the  divisor 
contains  no  letter  that  is  not  to  be  found  in  the  dividend  :  in 
the  second  place,  that  the  exponent  of  the  letters,  in  the  divi- 


2  3    -  +— a;3-2 X y3-2  x z^-^  =  -\-3xi/z. 


DIVISION.  39 

sor,  do  not  surpass  at  all  that  which  they  have  in  the  dividend  ; 
finally,  that  the  coefficient  of  the  divisor,  divides  exactly  that 
of  the  dividend. 

When  these  conditions  do  not  take  place,  then,  after  can- 
celling the  letters,  or  factors,  that  are  common  to  the  dividend 
and  divisor  ;  the  quotient  is  expressed  in  the  manner  of  a  frac- 
tion, as  in  (Art.  84).  • 

Ex.  5.  Divide  48a^b^c^d  by  Gia^^c^e. 

The  quotient  can  be  only  indicated  under  a  fractional  form, 
thus, 

48a^^c^d 
64a^Ve* 
But  the  coefficients  48  and  64  are  both  divisible  by  16,  sup- 
pressing this  common  factor,  the  coefficient  of  the  numerator 
will  become  3,  and  that  of  the  denominator  4.     The  letter  a 
having  the  same  exponent  3  in  both  terms  of  the  fraction,  it 
follows  that  d-^  is  a  common  factor  to  the  dividend  and  divisor, 
and  that  we  can  also  suppress  it.     The  exponent  of  the  letter 
b  is  greater  in  the   dividend  than  in  the   divisor  ;  it  is  ne- 
cessary to  divide  b^  by  6^,  and  the  quotient  will  be  b^^  or 
55 
—-=b^—^=b^f  which  factor  will  remain  in  the  numerator. 

With  respect  to  the  letter  c,  the  greater  power  of  it  is  in 

the  denominator  ;  dividing  c*  by  c^,  we  have  c^,  or  —=^c*~^ 

z=.r?^  therefore  the  factor  c^  will  remain  in  the  denominator. 

Finally,  the  letters  d  and  e  remain  in  their  respective  places  ; 
because,  in  the  present  state,  they  cannot  indicate  any  factor 
that  is  common  to  either  of  them. 

By  these  diflerent  operations,  the  quotient,  in  its  most  simple 

♦orm,  is  — — . 

'Note.     The  division  of  such  quantities  belongs,  properly 

speaking,  to  the  reduction  of  algebraic  fractions. 

Ex.  6.   Divide  "iGx^y^  by  9a?y.  Ans.  4yy. 

Ex.  7.  Divide  30a^Ay2  by  —(Sahy.  Ans.   —bay. 

Ex.  8.  Divide  —^^c-x^y  by  Ic^x^.  Ans.   —Gcxy. 

Ex.  9    Divide  — Aax'^y^  by  — axy"^,  Ans.   -{-^xy. 

Aa'^b"cx 
Ex.  10.  DWidiQ  \Qa^¥ ex  hy  —Aa^bdy.       Ans. 7 . 

EiP  11.  Divide  —  ISo^^V  by  \2a^¥x.        Ans.  —  — ^. 

Ex.  12.  Divide  llxyzw^  by  xzyw.  Ans.  11  w. 

Ex.  13.  Divide  —\'2aWc^  hy  —6abc.  Ans.  202^2^2. 


40  DIVISION. 


Ex.  14.  Divide  — ^x^xp-z^  by  x^y^'s^.  Ans. 2~2~2* 

Ex.  15.  Divide  39a9  by  ISa^.  Ans.  3a*. 

CASE  II. 

When  the  divisor  is  a  simple  quantity y  and  the  dividend  a  (?&m- 
pound  one. 

RULE. 

92.  Divide  each  term  of  the  dividend  separately  by  the 
simple  divisor,  as  in  the  preceding  case  ;  and  the  sum  of  the 
resuhing  quantities  will  be  the  quotient  required. 

Example  1.  Divide  \Sa^'\-2a^b-\-Qab'^  by  3a. 

18a3     ^  ,  ^d^b        ,        ^  6ai2 

Here, -=6a2, — a&,  and  ——=2*2  ; 

3a  3a  3a 

therefore,  ■8.^-+3a^i+6a^»^g^  ,_ 

3a 
Ex.  2.  Divide  20a2a;3  —  12a2a;2+8a3x2—2a*a;2  by  2aaj». 

20ff2j:3 

Here,  _— --=10aa:,  -12aV-r2aa;2=  —  Qa,  8aV-^2aa:2 
2aa3^ 

=  4  4a2,  and  '-2a^x'^^2ax'^=i  —  a^  ; 

20a2a;3— 12a2a:2  4-8a3a:2_2a*a;2      ,^ 

hence — =10aa: — 6a4-4a2— as^. 

2aa;2 

Ex.  3.  Divide  20a2a;  — 15aa;2-{-30axy2 — 5ax  by  5ax. 

Here  20a'^x-^5ax=4a,  —I5ax^-^5ax=  —3x,  SOaxy^-r- 

5ax  =  6y'^,  and  — 5ax-T-5ax=z  —  1  ; 

,        .         20a2x  — 15aa72-|-30aajy2_5aa;  „     .  ^  o     , 

therefore, -^ ^ =4a— 3a;  +  6y2_l. 

Oax 

Ex.  4.  Divide  5a^x—25a^x^+50a*x^  —  50a^x^-^25a^x^-~ 

5ax^  by  5aa;. 

Here  — —  =a5,  — =  — 5a%,  -^ =  +  lOa^^s 

oax  oax  oax 

-50a^x^  ^.  ,  3    +25a2x5  -Sarc^ 

— =  — lOa'^a?-^,  — =  -\-5ax*,  and  — =— x*  ; 

oaa?  oax  oax 

therefore,  a^— Sa^a^-l-lOa'-^x^ — 1  Oa^oj^  +  5aa;* — x^   is   the   quo- 
tient required. 

Ex.  5.  Divide  3a*a;2— 3a2a;*  by  — 3a2a;2.  Ans.  #— a'. 

Ex.  6    Divide  21a3a;3-7a2a;2-.14aa;  by  7 ax. 

Ans.  3a2a;^— ax--2. 


DIVISION.  41 

Ex.  7.  Divide   I2abc  —  ASax'^y^  +  64a26V  _  iQa-b^  by 

3c     3a:"v^ 

— 1 6a5.  Ans.  <z6  — -+  —r- 4abc^, 

4         0 

Ex.  8.  Divide  72a;2y222_i2aa;y3r4-245ca;y^  by  12a:yz. 

Ans.  6x1/2— a-\-2bG. 

Ex.  9.  Divide  4ir2y2— a?V+3aa;y  by  a:y. 

4        • 

Ans. xi/-{-3a. 

xy 

Ex.  10.  Divide5a-764-6c— 3flc2+9c3by  3c. 

Ans.5f-I^+2-ac+3c«. 
3c      3c 

Ex.  11.  Divide— 60a;"^y+50a?6y2_40a;5y3+30a;y—20a;3y« 

f  10a;2y6  — 5a:y^  by  —bxy. 

Ans.  y6_2a;y5  4.4a;2y*— 6xy+8a:*y2_i0a?SyH-12ar«. 

CASE  III. 

# 
When  the  dividend  and  divisor  are  both  compound  quantities, 

RULE. 

93.  Arrange  both  the  dividend  and  divisor  according  to  the 
exponents  of  the  same  letter,  beginning  with  the  highest^  and 
place  the  divisor  at  the  right  hand  of  the  dividend  ;  then  di- 
vide the  first  term  of  the  dividend  by  the  first  term  of  the  di- 
visor as  in  Case  I.,  and  place  the  result  under  the  divisor. 

Multiply  the  whole  divisor  by  this  partial  quotient,  and  sub- 
tract the  product  from  the  dividend,  and  the  remainder  will  be 
a  new  dividend. 

Again,  divide  that  term  of  the  new  dividend,  which  has  the 
highest  exponent,  by  the  first  term  of  the  divisor,  and  the  re- 
sult will  be  the  second  term  of  the  quotient.  Proceed  in  the 
same  manner  as  before,  repeating  the  operation  till  the  divi- 
dend is  exhausted,  and  nothing  remains,  as  in  common  arith- 
metic.     This  rule  is  evident  from  {Ait.  88). 

Example  1.  Divide  l2a^b^—6a^b^+Sa^^—4a^*^22a^+ 
5a''  by  4a2Z»2_2a3iH-5a*. 

It  can  be  readily  perceived  that  the  letter  a  is  the  one  to  be 
chosen,  in  order  to  arrange  the  terms  of  the  dividend  and  divi- 
sor according  to  its  powers,  beginning  with  the  dividend,  5a' 
is  the  term  which  contains  the  hi§|iest  pewer  of  a  ;  placing 
5a*'  for  the  first  term,  —22a^b,  for  the  second,  and  so  on  ;  the 
terms  of  the  dividend,  arranged  according  to  the  powers  of  a, 
are  written  thus ; 

5* 


42 


DIVISION. 


And  the  terms  of  the  divisor,  arranged  according  to  the  powers 
of  a,  are  written  thus  ; 

5a*— 2a3J+4a2J2, 


1      1 

10  ^o 

o  o 

a   a 

o»     o> 

<>-  O- 

+  + 

00  00 

a   a 

CJi        Ol 

+  4- 

1  1 

t— •  1— 1 

o  o 

►—  05 

^  ^ 

OS  a 

^.  ^ 

*r-  C> 

<U>      -J) 

^  i" 

1       1 

"  1 

►f*-  4^ 

»^ 

^ 

Cr-  O" 

Cr- 

*■     *. 

+  + 

+ 

00  00 

00 

a   a 

a 

^5     ^^ 

^3 

O  CN 

C?- 

o>    a> 

o> 

a  a 


t3 

60  60 

a  a 


+  + 


I 

60   •» 


Or 

a 

^§ 


+ 


The  sign  of  the  first  term  5a''  of  the  dividend  being  the 
same  as  that  of  5a*,  the  first  term  of  the  divisor,  the  sign  of 
the  first  term  of  the  quotient  is  +i  which  is  omitted  (Art.  14). 
Dividing  5a'  by  5a*,  the  quotient  is  a^,  which  is  written  under 
the  divisor.  Muhiplying  successively  the  three  terms  of  the 
divisor  by  the  first  term  a^  of  the  quotient,  and  writing  the 
product  under  the  corresponding  terms  of  the  dividend  ;  sub- 
tracting 5a'^ — 2a^b-\-4a^b^Apm  the  dividend,  the  remainder 
is 

—20a^+Sa^b^'-6a*P—4a^b*-^8a^^. 
Dividing  —20aH  the  first  term  of  this  new  dividend  by  5a\ 


DIVISION. 


43 


the  result  will  be  —Aa^h,  this  quotient  having  the  sign  — , 
because  the  dividend  and  divisor  have  different  signs 
Multiplying  all  the  terms  of  the  divisor  by  —Ad^b  ;  we  hav«» 
— 20a%-\-Sa^h'^ — IQa^b^ ;  subtracting  this  result  from  the  par* 
tial  dividend,  the  remainder  will  be  lOa^h^  —  Aa^b^-^-Sa^h^,  divid- 
ing the  first  term  of  this  new  partial  dividend,  1 0a*A2,  by  the  first 
term  5a*  of  the  divisor,  multiplying  all  the  divisor  by  the  result 
+2A^,  and  subtracting  the  product  from  the  last  partial  dividend, 
nothing  remains  ;  therefore  the  last  term  of  the  quotient  sought 
is  +26^,  and  the  entire  quotient  is  a^  —  Aa?'b-\-2b^. 

94.  It  is  very  proper  to  observe  that  in  division,  the  multi- 
plications of  different  terms  of  the  quotient  by  the  divisor, 
produce  frequently  terms  which  are  not  found  in  the  dividend, 
and  which  it  is  necessary  to  divide  afterward  by  the  first  term 
of  the  divisor.  These  terms  are  such  as  are  destroyed  when 
the  dividend  is  formed  by  the  multiplication  of  the  quotient 
Hnd  divisor. 

See  a  remarkable  example  of  these  reductions  : 

Ex.  2.  Divide  a^—b"^  by  a—b. 


Division. 


Dividend, 
a^-a^b 


a^b-- 

P 

a^b^ 

ab^ 

ab-^ 

-b^ 

ab'' 

-63 

Divisor, 
a-b 


Quotient. 
a^  +  ab  +  b^ 


Multiplication. 
Mul.  a  —  b 
by  a^-\-ab+b^ 


a^-a^b 
-{-a^b—ab^ 
-\-ab^- 


P 


-6' 


The  first  term  a^  of  the  dividend  divided  by  the  first  term 
a  of  the  divisor,  gives  a^  for  the  first  term  of  the  quotient ; 
multiplying  the  divisor  a  —  b  by  a^^  the  first  term  of  the  quotient, 
the  result  is  a^  —  a^  ;  subtracting  a^—a^b  from  the  dividend, 
the  term  a^  destroys  the  first  term  of  the  dividend  ;  but  there 
remains  the  term  ^-a^b,  which  is  not  found  at  first  in  the  divi- 
dend ;  therefore  the  remainder  is  a^b  —  b^.  Because  the  term 
a^b  contains  the  letter  a,  we  can  divide  it  by  the  first  term  of 
the  divisor,  and  we  obtain  -^-ab^  which  is  the  second  term  of 
the  quotient.  Multiplying  the  divisor  by  +ai,  the  product  is 
a^b-^ab"^,  which  being  subtracted  from  a^b—b^  ;  the  first  term 
a^b  destroys  the  term  a^b  which  arose  from  the  preceding 
operation  ;  but  there  remains  the  i€ifa  — oJ^,  which  being 
not  yet  in  the  dividend  ;  the  remainder  is  therefore  ab^—b^. 
Dividing  ab^  by  a,  the  result  is  b\  which  is  the  third  teim 


44 


DIVISION. 


of  the  quotient ;  multiplying  the  divisor  by  5^,  we  have 
ab^—P  ;  and  subtracting  this  result  from  the  last  remain- 
der, the  terms  of  both  destroy  one  another  ;  so  that  nothing 
remains. 

In  order  to  comprehend  well  the  mechanism  of  the  division, 
it  is  only  necessary  to  take  a  glance  at  the  multiplication  of 
the  quotient  a^-^ab-{-b^  by  the  divisor  a  —  b,  and  it  will  be  rea- 
dily seen  that  all  the  terms  reproduced  in  the  partial  divisions 
are  those  which  destroy  one  another  in  the  result  of  the  mul- 
tiplication. 


Ex.  3.  Divide  y^—l  by  y— 1 
Dividend. 

«/3       «/2 

Divisor. 
y-\ 

y    y 
y^-y 

Quotient. 

y- 
y- 

-1 
-1 

• 

Ex.  4.  Divide  a*— a;^  by  a—x. 

Dividend.  Divisor, 


a6-.x« 

a—x 

"■"  " "'"  ■■■'—                    • 
Quotient. 

a^x-x^ 
a'x—a^x^ 

a*a,2-a^6 
a^x^^a^x^ 

a^x^-x^ 
a^x^-a'x*- 

a^x*-x^ 
a^x*—ax^ 

ax'-x^ 

Ex.  5   Divide  sc^+a^  by  «+a. 


DIV 

Dividend. 
a:5  +  aa;* 

ISION 

.     Divisor. 

x-^-a 

Quotient. 
xi—ax^-\-a 

-ax^^a^ 
^ax^—a'^oc^ 

—a^x^-^-a^ 
—  a?x'^  —  a'^x 

45 


•a^oj+o* 


95.  When  we  apply  the  rule,  (Art.  93),  to  the  division  of 
algebraic  quantities  of  which  one  is  not  a  factor  of  the  other, 
we  know  it  is  impossible  to  effect  the  division  ;  because  that 
we  arrive,  in  the  course  of  the  operation,  at  a  remainder,  of 
which  the  first  term  cannot  be  divided  by  that  of  the  divisor. 
In  this  case,  the  remainder  is  made  the  numerator  of  a  frac- 
tion whose  denominator  is  the  divisor  ;  and  the  fraction  thus 
arising,  with  its  proper  sign,  is  annexed  to  the  other  part  of 
the  quotient,  in  order  to  render  its  value  complete. 

Ex.  6.  Divide  a'^-ya:^b^2h^  by  a^-^b'^. 

Dividend.  Divisor. 


a^  +  a^b^2h^ 
a^+ab"^ 


1st  rem. 


a?b—ab'^+2b^ 
a'b-^-b^ 


Quotient. 
a  +  b+ 


ab^ 


d'-\-b'^ 


2d  rem.  —ah'^-\-b'^ 

The  first  term  — a}p-  of  the  remainder,  cannot  be  divided  by 
c^,  the  first  term  of  the  divisor ;  thus  the  division  terminates 

at  this  point.     The  fraction  — -r^ — tj— ,  having  the  remainder 

for  its  numerator,  and  the  divisor  for  its  denominator,  is  an- 
nexed to  the  partial  quotient  a-\-b  ;  and  the  complete  quotient 

96.  It  is  necessary  to  remark,  that  the  operation  of  divi* 


46 


DIVISION. 


sion  may  be  considered  as  terminated,  when  the  highest  pow- 
er of  the  letter,  in  the  first  or  leading  term  of  the  remainder, 
by  which  the  process  is  regulated,  is  less  than  the  first  term 
of  the  divisor ;  as  the  succeeding  part  of  the  quotient,  after 
this,  would  necessarily  become  fractional ;  and  which  may  be 
carried  on,  ad  infinitum,  like  a  decimal  fraction. 

This  subject  belongs  to  algebraic  fractions,  and  as  it  is  of 
considerable  importance  in  analysis,  we  will  treat  of  it  with  a 
near  attenlion  in  the  next  Chapter. 

97.  In  the  preceding  examples,  the  product  of  the  first  term 
of  the  quotient  by  the  divisor,  is  placed  under  the  dividend  ; 
then  the  reduction  is  made  by  subtraction  ;  and  every  succeed- 
ing product  is  managed  in  like  manner.  In  the  following  ex- 
amples, the  signs  of  all  the  terms  of  the  product  are  changed 
in  placing  it  under  the  dividend  ;  and  then  the  reduction  is 
performed  by  the  rules  of  addition ;  which  is  the  method 
adopted  by  some  of  the  most  refined  Analysts. 

Ex.  7.  Divide  fl^+Sa^^'-^+i^—c*  by  aH  ^2+^2. 


Dividend. 
a^-\-2a^b'^-\-b^'-c^ 


1st.  rem. 


2d.  rem. 


a'JP-- 


,1r1 


+  6^. 


Divisor. 

Quotient. 
a2^b'^—c^ 


.aZb^—b^c^—b^ 


+  aV+52c2+c* 


Ex.  8.  Divide  6a;*— 96  by  3a:  — 6. 
Dividend.  Divisor. 


6x^  —  96 


+  12x3— 96 
—  12x3  +  24x2 


3x-6 

Quotient. 

2x3  +  4x2+8x+I6 


+24x2—96 
— 24x2+48x 


48x  — 96 
48x— 96 


DIVISION.  47 

Ex.  9.  DivideSa^— 4a352+4a3_j_2a3_52_|.iby2a3— 62+1. 


Dividend.  D 


ivisor. 


^a6_4a3^2_|_4a3_|.2a3_62^1 
•3a6-f4a362_4a3 


2a3_j2  4.i 
.2a3 -1-62—1 


203-62+1 

Quotient. 
4a3  +  l 


98.  The  division  of  algebraic  quantities  can  be  sometimes 
facilitated  by  decomposing,  at  sight,  a  quantity  into  its  fac- 
tors ;  thus,  in  the.  above  example,  the  divisor  forms  the  last 
three  terms  of  the  dividend,  it  is  only  necessary  to  seek  if  it 
be  a  factor  of  the  first  three  ;  but  those  have  visibly  for  a 
common  factor  4a^,  for  8a'^-'4aW-i-4a^  =  4a^x(2a^—b^i-l), 

By  this  observation,  the  dividend  will  become 

or       (2a3--62-f-I)x(4a3+l): 
therefore  the  division  is  immediately  effected,  by  suppressing 
the  factor  2a^—b^-\-l  equal  to  the  divisor,  and  the  quotient 
will  be  4a3  +  l. 

Experience,  in  algebraic  calculations,  will  suggest  a  great 
many  remarks  of  this  kind,  by  which  the  operations  can  be 
frequently  abridged. 

99.  It  sometimes  happens  that,  in  arranging  the  dividend 
and  the  divisor  according  to  the  same  letter,  there  occur  seve- 
ral terms  in  which  this  letter  has  the  same  exponent :  In  this 
case,  it  is  necessary  to  range  in  the  same  column  those  terms, 
observing  to  order  them  according  to  another  letter,  common 
to  the  two  quantities. 

Ex.  10.  Divide  -.a*h^^b^c*— 0^0^—0^+ 2a*(^i-b^+2Mc^ 
■\^a^^  by  a'^—b^  —  c\ 

Ordering  the  dividend  according  to  the  letter  a,  we  will 
place  in  the  same  column  the  terms  — a'^b^  and  +2a'*c2,  in 
another  the  terms  +0^6*  and  —a^c*  ;  finally,  in  the  last 
column  t^je  three  terms  +6^,  -{-2b*c^,  +6^6'*,  ordering  them 
according  to  the  exponents  of  the  letter  6 ;  then  the  quanti- 
ties, so  arranged,  will  stand  thus  : 


18  DIVISION. 


Dividend.  Divisor. 


1st  rem.  —2a'b"+ar-h^+¥ 

4-   a4c2— a2c44-2Mc2 
4-  62c4 
-f-2a*J2_2a2M 
— 2a262c2 


2_i2_c2 


Quotie7it. 

—  d^—2a^b'^—¥ 


2d  rem.  +a^c'^—  a^b^    -\-b^ 

—2aH^c^  +  2b^c^ 


3d  rem.  — a^^*    +56 

— a2^>2c2+2Mc2 

+  &2c4 
+  ^254      _J6 

-^4^2 


4th  rem.  ^a^^c^-^b^c^ 

+  b^c* 
J^aWc^—b^c^ 

-&2c4 


*  * 

Ex.  II.  Divide  ffx*  -  (5+ac)a;3+(c4-Jc+a)a;2—(c2+5)a: 
4-c  by  ax^ — bx-\rc. 

Dividend.  Divisor. 


ax*—{b  +  ac)x^  +  {c-\-bc-\-a)x'^-'{c^+b)x-{-c 
-ax*  -\-bx^  —  ca;2 

—acx^-\-(bc-{-a)x^'^{c'^-\-b)x+c 
■i-acx^  — bcx"^         -i-c^x 


ax^ — bx-\-c 

Quotient. 
a?2 — cx-{-i 


ax^ — bx-\-c 
•ax^-{-bx — c 


DIVISION.  49 

100.  The  following  practical  examples  maybe  wrought  ac- 
cording to  either  of  the  methods  pointed  out,  (Art.  93,  97) ; 
but  in  complicated  cases,  the  latter  should  be  preferred.  See 
Example  10. 

Ex.  12.  Divide  x^—x^-{-x^—x^-[-2x^l  by  x'^+x.  —  l. 

Ans.  oc^—x^-^-x^ — x-f  1. 

Ex.   13.  Divide  a^  +  5a*x  +  l0a^x'^—l0a'2x^-\-5ax*—x^  by 

a^—Sa'^x+Sax^—x^.  • 

Ans.  a^ — 2ax-\-x^, 
Ex.  14.  Divide  2x3— 19«2H-26a;-16  by  a:-8. 

Ans.  2x^—3x-j-2' 
Ex.  15.  Divide  483^3— 76ay2_64a2yH-105a3  by  2y  — 3a. 

Ans.  24y2_2ay  — 35a2. 
Ex.   16.  Divide  a2— 6^  by  a— ^.  Ans.  a+5. 

Ex.   17.  Divide  a*— a:*  by  a^—x"^,  Ans.  a^-\-x^. 

Ex.  18.  Divide  a^  —  b^  by  a^+2a'^b  +  2ab^-\-b^. 

Ans.  a^—2a^b-\-2ab^-~b\ 
Ex.  19.  Divide  a^+a^b'^+b^  by  a^—ab~\-b'^. 

Ans.  a^i-ab+b\ 
Ex.  20.  Divide  25a;6— a;*— 2x3— 8a;2  by  5x3  —  4x2. 

Ans.  5x3+4x2+3x+3. 
Ex.  21.  Divide  a^+4ab-{-4b-+c^  by  a+2b. 

An8.a-{-2b  +  -^ 

a-{-2o 

Ex.  22.  Divide8a*— 2^35  — 130=^^-30 J3  by  4a2+5ai4- 62. 

Ans.  2a^—3ab. 
Ex.  23.  Divide  20a5—41a464-50a3i2__45a2i3^25a6*—66« 
by  ta^  —  5ab-\-2b^. 

Ans.  5a3_4a2J^_5aJ2_3j3. 

Ex.  24.  Divide  a*  +  8a3x+24a2a;2-j-32ax3+16x*  by  a-f2x. 

0  Ans.  a3_|_6a2j:-f-l2ax2+8x3. 

E^  25.  Divide  x4—(«—J)x3_|_(^_^^,^3j^2_^(^^__3^)^ 

4-3J9  by  x2— ax+p.  Ans.  x24-6x4-3. 

Ex.  26.  Divide  ax3-(a2_j_j)a;24.^2by  ffa;— 6. 

Ans.  x'^—ax—b. 
Ex.  27.  Divide  y6+«2y4^5Y2_^6_2J2y4_a4y2_2a462_ 
a26*  by  y*4-2a2y2_|_a4_^>2y2_^^2^2^ 

Ans.  y2 — ^2- Z>2, 
Ex.  28.  Divide  9x6  — 46x^  +  95x2+ 150x  by  x2—4x~5. 

Ans.  9x4  — 10x3.4-5x2— 30x. 
Ex.  29.  Divide  6a'^-\-9a'^—l5a  by  3^2— 3a. 

Ans.  2a2 4.2^4- 5. 
Ex.  30.  Divide 2a*— 16a36+31a2i2_38a63^246*by2a2_- 
3^+4^^.  Ans.  a^-^5ab-{-6b\ 

6 


50  GENERAL  THEOREMS. 

Ex.  31.  Divide  a^  +  Sa^x  +  2Sa^x^  +  SGa^ar^  -f  70a^x^  + 
56a^x^+28a'^x^-\-8ax'^ -^-x^  by  a^+4a^x-\-ea'^x'^-\-iax^-\-x^. 

Ans.  a*-\-4a^x+6a^x^-{-4ax^-{-x*. 

Ex.  32.  Divide  ««— Ga'^ac-f  15a*a;2— 20a^a'3+i5a2a;4_6aa:S 
-\-x^  by  a^ — 3o%+3aa:2— a:^.  ^j^g  a^^Sa^x-\-3ax^—x^ 

^  V.  iSomc  General  Theorems ^  Observations,  &c. 

|»  101.  Newton  calls  Algebra  Universal  AritJimetie.  This 
denomination,  says  Lagrange,  in  his  Traite  de  la  Resolution 
des  Equations  numeriques,  is  exact  in  some  respects  ;  but  it 
does  not  make  sufficiently  known  the  real  difference  between 
Arithmetic  and  Algebra. 

Al^gebra  differs  from  Arithmetic  chiefly  in  this  ;  that  in  the 
latter,  every  figure  has  a  determinate  and  individual  value 
peculiar  to  itself;  whereas  the  algebraic  characters  being  ge- 
neral, or  independent  of  any  particular  or  partial  signification, 
represent  all  sorts  of  numbers,  or  quantities  according  to  the 
nature  of  the  question  to  which  they  are  applied. 

Hence,  when  any  of  the  operations  of  addition,  subtraction, 
&€.,  are  to  be  made  upon  numbers,  or  other  magnitudes,  which 
are  represented  by  the  letters,  a,  h,  c,  &c.,  it  is  obvious  that  the 
results  so  obtained  will  be  general ;  and  that  any  particular 
case,  of  a  similar  kind,  may  be  readily  derived  from  them,  by 
barely  substituting  for  every  letter  its  real  numeral  value,  and 
then  computing  the  an^unt  accordingly. 

Another  advantageUllso,  which  arises  from  this  general 
mode  of  notation,  is,  that  while  the  figures  employed  in  Arith- 
metic disappear  in  the  course  of  the  operation,  the  characters 
used  in  Algebra  always  retain  their  original  form,  so  as  to 
show  the  dependence  they  have  upon  each  other  in  every 
part  of  the  process  ;  which  circumstance,  together  with  that 
of  representing  the  operations  of  add-on,  subtraction,  &c.,  by 
means  of  certain  signs,  renders  both  the  language  and  al^rithm 
of  this  science  extremely  simple  and  commodious. 

Besides  the  advantages  which  the  algebraic  method  of  no- 
tation possesses  over  that  of  numbers,  it  may  be  observed,  that 
even  in  this  early  part  of  the  science  we  are  furnished  with 
the  means  of  obtaining  several  general  theorems  that  could 
not  be  well  established  by  the  principles  of  Arithmetic. 

102.  The  greater  of  any  two  numhers  is  equal  to  half  their  sum 
added  to'  half  their  difference,  and  the  less  is  equal  to  half  their 
sum  minus  half  their  difference. 

Let  a  and  b  be  any  two  numbers,  of  which  a  is  the  greater ;  let 
their  sum  be  represented  by  s ;  and  their  difference  by  d.  Then, 


GENERAL  THEOREMS.  51 

a-\-b=sl 
a^b=dS 


•  .*.  by  addition,  2a:=s-\-d    (Art.  48) ; 

s      d 
and  a—~-\--     (Art.  51) 

By  subtraction,  2&=*— t/     (Art.  49) ; 
s     d 


2  •  2     ^ "> 

;| 


and         .-.  *  =  o~o    C^'^-  ^0 

2     •<& 

*Cor."l.  Hence  if  tbe  sum  and  difference  of  any  two  nUm 

bers  be  given,  we  can  readily  find  each  of  the  numbers  ;  thus, 

if  s  be  equal  to  the  sum  of  two  numbers,  and  d  equal  to  the 

s-^-d 
difference ;  then  the  general  expression  for  the  first,  is  — — - , 

2 

and  for  the  second 


2 

Whatever  may  be  the  numeral  values  that  we  assign  to  s 
and  d^  or  whatever  values  these  letters  must  represent  in  a 
particular  question,  we  have  but  to  substitute  them  in  the  above 
expressions,  ia  order  to  ascertain  the  numbers  required :  For 
example. 

Given  the  sum  of  two  numbers  equal  to  36,  and  the  diffe- 
*rence  equal  to  8  : 

Then,  by  substituting  36  for  ^,  and  8  for  d,  in  — - —  and 

s~d  ^        s-\-d      36-f8     44     „^       ^  s-^d     36—8 

,  we  nave  = — - — =-—=22,  and 


2  2  '  2  2 

2ft 

—-=14.     So  that,  22  and  14  are  the  numbers  required. 

Cor.  2.  Also,  if  it  were  required  to  divide  the  number  s 
into  two  such  parts,  that  the  Jirst  will  exceed  the  second  by  d. 
It  appears  evident,  that  the  general   expression  for  the  first 

P     I       /7  •  g fi 

part  is  — - — ,  and  for  the  second  — - — ;  s  and  d  representing 
2  2 

any  numbers  whatever. 

s-\-d 
103.  The  general  expression  — - —  maybe  found  after  the 

2 

manner  of  Gamier.     Thus,  let  x  represent  the  first  part ;  then 

according  to  the  enunciation  of  the  question,  x—d  will  be  the 

second ;  and,  as  any  quantity  is  equal  to  the  sura  of  all  its 

parts,  we  have  therefore, 

x-\-x — dzzzSj  or  2x — d:=:$. 


52  GENERAL  THEOREMS. 

This  equality  will  not  be  altered,  by  adding  the  number  d 
to  each  member,  and  then  it  becomes, 

2x--d+d=s-\-d,  or  2x=zs+d; 

dividing  each  member  by  2,  we  have  the  equality,  S=  ; 

in  which  we  read  that  the  number  sought  is  equal  to  half  the 
sum  of  the  two  numbers  s  and  d ;  thus  the  relation  between 
the  unknown  and  known  numbers  remaining  the  same,  the 
question  is  resolved  in  general  for  all  numbers  s  and  d. 

104.  We  have  not  here  the  numerical  value  of  the  unl^nown 
quantity  ;  but  the  system  of  operations  that  is  to  be  performed 
upon  the  given  quantities  ;  in  order  to  deduce  from  them,  ac- 
cording to  the  conditions  of  the  problem,  the  value  of  the  quan- 
tity sought ;  and  the  expression  that  indicates  these  opera- 
tions, is  called  a  formula. 

It  is  thus,  for  example,  that  if  we  denote  by  a  the  tens  of  a 
number,  and  the  units  by  b,  we  have  this  constant  composi- 
tion of  a  square,  or  this  formula, 

a'^-{-2abfh^  ; 
this  algebraic  expression  is  a  brief  enunciation  of  the  rules  ta 
be  pursued  in  order  to  pass  from  a  number  to  its  square. 

105.  From  whence  we  infer  that,  if  a  number  be  divided  into 
any  two  parts,  the  square  of  the  number  is  equal  to  the  square  of 
the  two  parts,  together  with  twice  the  product  of  those  parts. 

Which  may  be  demonstrated  thus  ;  let  the  number  n  be  di- 
vided into  any  two  parts  a  and  b  ; 

Then  n=:a-\-b, 
and      n=a-\-b; 


.-.by  Muhiplication,  n'^=a'^-{-2ab  +  b^  (Art.  50). 
106.  If  the  sum  and  difference  of  any  two  numbers  or  quan- 
tities be  multiplied  together^  their  product  gives  the  difference 
of  their  squares,  observing  to  take  with  the  sign  —  that  of 
.    the  two  squares  whose  root  is  subtracted.^ 

Let  M  and  n  represent  any  two  quantities,  or  polynomials 
whatever,  of  which  m  is  the  greater;  then  (m  +  n)x(m  — n) 
is  equal  to  m^-— n^  ;  for  the  operation  stands  thus  ; 

(M  +  N)x(M~N)=:rM''+MN  J    =^2^^2 

—  MN— N^  S 

107.  When  we  put  M  =  a^,  and  ^=zb^;  then, 

(a3  4-i3)x(«3_j3)_^6_^,6  .  (See  Ex.  9.  page  30). 

Where  a^  is  the  square  of  a^,  and  b^  that  of  b^,  and  this  last 
square  is  subtracted  from  the  first. 

Reciprocally,  the  difference  of  two  squares  M^ — N^,  can  he 
put  under  the  form  (m-|-n)x(m— n). 


GENERAL  THEOREMS.  53 

This  result  is  ^formula  that  should  be  remembered. 

108.  The  difference  of  any  two  equal  powers  of  different  quanti- 
ties is  always  divisible  by  the  difference  of  their  roots,  whether 
the  exponent  of  the  power  he  even  or  odd.     For  since 

x^ — a^ 

=a;-f  a; 

X — a 

x^ — a'^  ^^ 

=a;2-f  flta?4-a^  ;  ^k 

X — a  ^ 

aj* — (1% 

—x^-\-ax'^-\'a'^x-\'a^\ 

X — a 

=^x^-{-ax^-\-a'^x'^-{-a^x-\-a^  \  ( 


X — a 
x^  —  a^ 


■.x^-\-ax^-\-a^x^-\-a^x'^-{-a^x-\-a^ ; 


x—a 

We  may  conclude  that  in  general,  a;'"  — a*»  is  divisible  by  a:— a, 
m  being  an  entire  positive  number  ;  that  is, 

X — a 

109.  The  difference  of  any  two  equal  powers  of  different  quanti' 
ties,  is  also  divisible  by  the  sum  of  their  roots,  tohen  the  expo- 
nent of  the  power  is  an  even  number.     For  since 

x'^—a^ 
- — =x—a; 
x-{-a 

=  a;^ — ax'^-\-a'^x — a^  ; 

x-^a 

&c.        &LC 

Hence  we  may  conclude  that,  in  general, 


.a:2m-i_aa;-'"-2.4-  .  .  -fa2»«-2a:—a2'«-i  .  (2). 


x-\-a 

110.  And  the  sum  of  any  two  equal  powers  of  different  quanti- 
ties, is  also  divisible  by  the  sum  of  their  roots,  when  the  expo* 
nent  of  the  power  is  an  odd  number.     For  since 


x-\-a 


z=x'^—ax^-{-a'^x'^-\-a^x+c^ 


x-\-a 
Hence  we  may  conclude  that,  in  general, 

ffl!!d!lf!!;!l=:a;2m_oa,2„-i+.  .-02»u-la;4-a2«.  (3). 
x-\-a 

6* 


54  GENERAL  THEOREMS. 

111.  In  the  formulae  (1),  (2),  (3),  as  well  as  in  all  others  of 
a  similar  kind,  it  is  to  be  observed,  that  if  m  be  any  whole  num- 
ber whatever,  '2m  will  always  be  an  even  number,  and  2m -fl 
an  odd  number  ;  so  that  ^m  is  a  general  formula  for  even  num- 
bers, and  2m-\-\  for  odd  numbers. 

112.  Also,  if  a  in  each  of  the  above  formulae,  be  taken  =1, 
and  X  being  always  considered  greater  than  a  \  they  will  stand 
as  follows  : 

-—x'^'^-\-x'^-^^x^-^-\- +a:+l  .  .  .  (4). 

X — 1  ^  ' 

f_!!_Z__a;2m_l_a,2m    2j^^1m-Z^  .  .  .  +3?— 1    .  .  .  (5). 

_^-a;2m_a;2m_i_|.a,2m-2_,  ,  .— a;+l  .  .  .  (6). 

113.  x\nd  if  any  two  unequal  powers  of  the  same  root  be 
taken,  it  is  plain,  from  what  is  here  shown,  that 

x^—xn^  or  a;"(a;'"-"— 1) (7), 

is  divisible  by  rr— 1,  whether  m—n  be  even  or  odd ;  and  that 

a;"»— a;",  or  a;"(a;'"-"— 1) (8), 

is  divisible  by  oj  +  l,  where  m—n  is  ai^ven  number  ;  as  also 
that 

a">+a;'*,  or  a:"(a:'«~"+l) (9), 

is  divisible  by  a:-{-l,  when  m — n  is  an  odd  number. 

114.  It  is  very  proper  to  remark,  that  the  number  of  all 
the  factors,  both  equal  and  unequal,  which  enter  in  the  for- 
mation of  any  product  whatever,  is  called  the  degree  of  that 
product.  The  product  a^i^c,  for  example,  which  comprehends 
six  simple  factors,  is  of  the  sixth  degree;  this,  d}}p-c  is  of  the 
tenth  degree  ;  and  so  on. 

Also,  that  if  all  the  terms  of  a  polynomial,  or  compound 
quantity,  be  of  the  same  degree,  it  is  said  to  be  homogeneous. 
And  it  is  evident  from  the  rules  established  in  Multiplication. 
that  if  two  polynomials  be  homogeneous  ;  their  product  will  be 
also  homogeneous ;  and  of  the  degree  marked  by  the  sum  of  the 
numbers  which  desigjiate  the  degree  of  those  factors. 

Thus,  in  Ex.  1,  page  29,  the  multiplicand  is  of  the  fourth 
degree,  the  multiplier  of  the  third,  and  the  product  of  the  de- 
gree 4  +  3,  or  of  the  seventh  degree. 

In  Ex,  12,  page  31,  the  multiplicand  is  of  the  third  degree, 
the  multiplier  of  the  third,  and  the  prochict  of  the  degree  3+3, 
or  of  the  sixth  degree. 

Hence,  we  can  readily  discover,  by  inspection  only,  the  er- 
rors of  a  product,  which  might  be  committed  by  forgetting 
some  one  of  the  factors  in  the  partial  multiplications. 


CHAPTER  II. 

ON 

ALGEBRAIC  FRACTIONS. 

115.  We  have  seen  in  the  division  of  two  simple  qtHntities 
(Art.  84,)  that  when  certain  letters,  factors  in  the  divisor,  are  not 
common  to  the  dividend^  and  reciprocally,  the  division  can  only 
be  indicated,  and  then  the  quotient  is  represented  by  a  fraction 
whose  numerator  is  the  product  of  all  the  letters  of  the  dividend, 
not  common  to  the  divisor,  and  denominator,  all  those  letters  of 
the  divisor,  not  common  to  the  dividend. 

Let,  for  example,  abmn  be  divided  by  cdmn ;  then 
ahmn     ah 
cdmn     cd' 

It  may  be  observed,  that  the  fraction  —z  may  be  a  whole 

number  for  certain  numeral  values  of. the  letters  a,  b,  c,  and  d ; 
thus,  if  we^  had  a  =  4,  1  =  6^  c=i2,  d=z3  ;  but  that,  generally 
speaking,  it  will  be  a  numerical  fraction  which  can  be  reduced 
to  a  more  simple  expression. 

^  I.   Theory  of  Algebraic  Fractions. 

116.  It  is  evident  (Art.  103,)  that  if  we  perform  the  same  opera' 
tion  on  each  of  the  two  members  of  an  equality^  that  is,  upon 
two  equivalent  quantities  or  numbers,  the  results  shall  always 
be  equal. 

It  is  by  passing  thus  from  the  fractional  notation  to  the  al- 
gorithm of  equality,  that  the  process  to  be  pursued  in  the 
researches  of  properties  and  rules,  becomes  simple  and  uni- 
form, n 

117.  Let  therefore  the  equality  be 

a=bxv (1). 

when  we  divide  both  sides  b«||  which  has  no  factor  common 


with  a,  we  shall  have 


i=" (2)- 


Thus  V  will  represent  the  value  of  the  fraction  -r,  or  the  quo- 
tient of  the  division  of  a  by  5. 


56  ALGEBRAIC  FRACTIONS. 

lis.  If  the  numerator  and  denominator  of  a  fraction  be  both  mul- 
tiplied^ or  both  divided  by  the  same  quantity ^  its  value  will  not 
be  altered. 

For,  if  we  multiply  by  m  the  two  members  of  the  equality 
(1),  we  will  have  these  equivalent  results, 

ma=mbxv (3) ; 

dividing  both  by  mb,  we  shall  have 
•  ma 

but  T=v  >  therefore 

0 

ma  a  . 

mb=''=b ('''• 

m  being  any  whole  or  fractional  number  whatever. 

119.  If  the  fraction  is  to  be  multiplied  by  m,  it  is  the  same  whether 
the  numerator  be  mulliplied  by  it,  or  the  denominator  divided 
by  it. 

For,  if  we  divide  by  b,  the  two  members  of  the  equality  (3), 

we  obtain  the  following, 

ma  .. 

-^=mXv (5). 

The  equality  (1)  may  also  be  put  under  the  form 

a=-  bxmv (6), 

m 

whence  we  derive,  dividing  each  side  by  -i, 
T^='wXv (7). 

m 

120.  If  a  fraction  is.  to  be  divided  by  m,  it  is  the  same  whether  the 
numerator  be  divided  by  m,  or  the  denomi?iator  mulliplied  by  it. 
For,  from  the  equality  (1),  we  deduce  these 

fl) -=bx-,a=mbx- (9), 

^  '  mm  m 

dividing  the  first  by  b  and  the  second  by  mbj  in  order  to  have 

— ,  they  become  ^ 

m 

(10)....==^;^-!;... .(11). 
^     '  b      m     mb     m  ^     ' 

o 

It  is  to  be  observed,  that  in  t,  the  numerator  is  —  and  the 


ALGEBRAIC  FRACTIONS.  57 

denominator  b,  and  that  we  employ  the  greater  line  for  se- 
parating the  numerator  from  the  denominator. 

121.  If  two  fractions  have  a  common  denominator,  their  sum 
will  be  equal  to  the  sum  of  their  numerators  divided  by  the 
common  denominator. 

For,  let  now  the  two  equalities  be 

(12) a=bXv\  a=bxv' (13), 

corresponding  to  the  fractions 

a         a'        , 

which  have  the  same  denominator  ;  adding  the  two  equalitio 
(12)  and  (13),  we  shall  have 

a-{-a  z:^bv-\-bv  ^^b^v-^-W)  ; 
and  dividing  both  members  by  6,  in  order  to  have  the  sum 
sought  v\-v\  it  becomes 

—r—^v-Yv'  ....  (14). 

Note.  In  adding  the  above  equalities,  the  corresponding 
members  are  added ;  that  is,  the  two  members  on  the  left- 
hand  side  of  the  sign  =,  are  added  together,  and  likewise 
those  on  the  right.  The  same  thing  is  to  be  understood  when 
two  equalities  are  subtracted,  multiplied,  &;c. 

122.  If  two  fractions  have  a  common  denominator,  their  differ* 
ence  is  equal  to  the  difference  of  their  numerators  divided  by 

^  the  Common  denominator. 

For,  if  we  subtract  the  equality  (13)  from  (12),  we  shall  have 
a  —  a=bv  —  bv''  =  b{v — v^)  ; 
dividing  each  side  by  b,  and  we  will  obtain 
a — a' 
-J-^"-" (15)- 

123  Let  us  suppose  that  the  fractions  have  different  de- 
nominators, or  that  we  have  the  equalities 

azzzb  .  V,  a'=zb'  .  v'  ; 
we  will  multiply  the  two  members  of  the  first  by  b\  and  those 
of  the  second  by  b,  an  operation  which  will  give 

ab'  =  byv,  a'bzzzhb'v'  ; 
then  adding  and  subtracting,  we  have 

ab'^ab  —  hb'(v^v'),  ^-''^  "^r^^-^.a^ 

the  double  sign  JL  which  we  led^A  plus  or  7wmt/.y,  indicating  at  '^^ 
the  same  time  both  addition  and  subtraction  ;  dividing  eachy 
side  by  bb\  in  order  to  find  the  sum  and  difference  sought 
v-^v'j  we  will  have  .^      .,  i- >« 


58  ALGEBRAIC  FRACTIONS. 

from  whence  we  might  readily  derive  the  rule  for  the  addi- 
tion and  subtraction  of  fractions  not  reduced  to  the  same  de- 
nominator. 

124.  It  would  be  without  doubt  more  simple  to  have  re- 
course to  property  (4)  in  order  to  reduce  to  the  same  denomi- 
ivator  the  fractions 

a         a' 

but  our  object  is  to  show,  that  the  principle  of  equality  is  suf- 
ficient to  establish  all  the  doctrine  of  fractions. 

125.  We  have  given  the  rule  for  multiplying  a  fraction  by 
a  whole  number,  which  will  aho  answer  for  the  multiplication 
of  a  whole  number  by  a  fraction. 

Now,  let  us  suppose  that  two  fractions  are  to  be  multiplied 
by  one  another. 

Let  the  two  equalities  be 

a=zb  .  V,  a'=:zb'  .  v^ ; 
multiplying  one  by  the  other,  the  two  products  will  be  equal ; 
thus, 

aa^=by  .  vv\ 
and  dividing  each  side  by  bb',  in  order  to  have  the  product 
sought  vv\  we  will  obtain 

i-' 0^);      .         ... 

Therefore  the  product  of  two  fractions,  is  a  fraction  having 
for  its  numerator  the  product  of  the  numerators,  and  for  its  de- 
nominator  that  of  the  denominators. 

126.  It  now  remains  to  show  "how  a  whole  number  is  to  be 
divided  by  a  fraction  ;  and  also,  how  one  fraction  is  to  be  di- 
vided by  another. 

Let,  in  the  first  case,  the  two  equalities  be 
m=^m  ;  a=zb  .  v  ; 
if  we  divide  one  by  the  other,  the  two  quotients  will  be  equal, 
that  is, 

m     m 
a       bv ' 
and  multiplying  both  sides  by  b,  in  order  to  have  the  expres- 


m 


sion  — ,  we  shall  find 


™*=^ (18). 

a       V 


ALGEBRAIC  FRACTIONS.       n  59 

Therefore,  to  divide  a  whole  number  by  a  fraction,  we  must 
multiply  the  whole  number  by  the  reciprocal  of  the  fraction,  or 
which  is  the  sa/rie,  by  the  fraction  inverted 

Let,  in  the  second  case,  the  two  equalities  be 
a=:b  .  V,  a'  —b'  .  v' ; 
if  the  first  equality  be  divided  by  the  second,  we  shall  have 
a      b  .  V 

multiplying  each  side  by  y  and  dividing  by  b,  for  the  purpose 
of  obtaining  the  expression  — „  we  will  arrive  at 

"^^Cjx*: (19).  • 

ab     V      b     a'  ^     ' 

Therefore,  to  divide  one  fraction  by  another,  we  must  multiply 
the  fractional  dividend  by  the  reciprocal  of  the  fractional  divisor^ 
or  which  is  the  same,  by  the  fractional  divisor  inverted. 

127.  These  properties  and  rules  should  still  take  place  in 
case  that  a  and  b  would  represent  any  polynomials  whatever. 

Accordincr  to    the    transformation  a-^-=—r,   demonstrated 

(Art.  86),  we  can  change  a  quantity  from  a  fractional  form  to 
that  of  an  integral  one,  and  reciprocally.     So  that,  we  have 

-=bx-=bXar^  —ba-"^,  -^=bx-T=h  X  a-^  z=z  ba-^ ,  and 
a  a  a^  a"- 

a-2i-2^-2--      X-75X-75— -77?i5-     In  like  manner  any  quan- 
(P-      b^      d^     a^b^d^ 

tity  may  be  transferred  from  the  numerator  to  the  denominator, 

and  reciprocally,  by  changing  the  sign  of  its  index  : 

„,        cP'b         b         bc"^ c-2  a-'^x~'^z-^ c'^y^ 

'  c^  ~~  a-'^c'^  ~  a-2      a-'^b-^ '  c-^b'^y-^        d^b'^x^z 

128.  If  the  signs  of  both  the  numerator  and  denominator  of  a 
fraction  be  changed,  its  value  will  not  be  altered. 

,         — a      -\-a         a     a     a  —  b     b — a 


b     +6  b~'b'  c  —  d     d—c 

Which  appears  evident  from  the  Division  of  algebraic  quan- 
tities having  like  or  unlike  signs.  Also,  if  a  fraction  have  the 
negative  sign  before  it,  the  value  of  the  fraction  will  not  be  altered 
by  making  the  numerator  only  negative,  or  by  changing  the  signs 
of  all  its  terms. 


60  ALGEBRAIC  FRACTIONS. 

Thus,  -_=+  and -_=+-_=__. 

0  0  c-\-d         c-f-d      c  +  rf 

And,  in  like  manner,  the  value  of  a  fraction  having  a  negative 
sign  before  it,  will  not  be  altered  by  making  the  denominator 
only  negative :  Thus, 

a — b  a — b a — b 

c — d  d — c      d — c* 

129.  Note.  It  may  be  observed,  that  if  the  numerator  be 
equal  to  the  denominator,  the  fraction  is  equal  to  unity  ;  thus, 

if  az=bj  thenT=-=l  :  Also,  if  a  is  >5,  the  fraction  is  great 

er  than  unity  ;  and  in  each  of  those  two  cases  it  is  called  an 
improper  fraction :  But  if  a  is  <^b,  then  the  fraction  is  less  than 
unity,  and  in  this  case,  it  is  called  a  proper  fraction. 

§  II.  Method  of  finding  the  Greatest  Common  Divisor  of  two  of 
more  Quantities. 

130.  The  greatest  common  divisor  of  two  or  more  quanti- 
ties, is  the  greatest  quantity  which  divides  each  of  them  ex- 
actly. Thus,  the  greatest  common  divisor  of  the  quantities 
16^2^2^  \2a^bc  and  4abc^,  is  4ab. 

131.  If  one  quantity  measure  two  others,  it  will  also  mea- 
sure their  sum  or  difference.  Let  c  measure  a  by  the  units  in 
m,  and  b  by  the  units  in  n,  then  a=:mcj  and  b=nc;  therefore 
a-i-b=zmc-\-nc=:(m-\-n)c;  and  a — b^=mc—nc={m—n)c;  or 
a±6=(m±n)c  ;  consequently  c  measures  a-{-b  (their  sum) 
by  the  units  in  m+n,  and  a—b  (their  difference)  by  the  units 
in  m — n. 

132.  Let  a  and  b  be  any  two  numbers  or  quantities,  where- 
of a  is  the  greater  ;  and  let  p=  quotient  of  a  divided  by  b,  and 
c=  remainder  ;  q=^  quotient  of  b  divided  by  c,  and  d=  re- 
mainder ;  r—  quotient  of  c  divided  by  d^  and  the  remainder  —0  ; 
thus, 

b)a{p 
pb 


c)b(q 
qc 

d)c(r 
rd 


Then,  since  in  each  case  the  divisor  multi- 
plied by  the  quotient  joZ?/^  the  remainder  is  equal 
to  be  dividend  ;  we  have 
c=rd,  hence  qc—qrd  (Art.  50) ; 
*b=qc-\-d=qrd-\-d={qr-\-\)d  ;    and  pb=pqrd 
-\-pd  =  {pqr-\~p)d  (Art.  61.)  , 
a=pb-\-  c=pqrd'\-pd-\-rd=.{pqr-\-p-k-r)d. 


ALGEBRAIC  FRACTIONS.  61 

Hence,  since  p,  q,  and  r,  are  whole  numbers  or  integral 
quantities^  d  is  contained  in  b  as  many  times  as  there  are 
units  in  qr-\-l,  and  in  a  as  many  times  as  there  are  units  in 
pqr-[-p-{-r  \  consequently  the  last  divisor  d  is  a  common 
measure  of  a  and  b ;  and  this  is  evidently  the  case,  whatever 
be  the  length  of  the  operation,  provided  that  it  be  carried  on 
till  the  remainder  is  nothing. 

This  last  divisor  d  is  also  the  greatest  common  measure  of 
a  and  b.  For  let  a?  be  a  common  measure  of  a  and  b  \  such 
that  a=mx,  and  b7=nx,  then  pb=:pnx  ;  and  Cz=a — pb=mx — 
pnx=(m—pn)x,  also  dz=b~qc=.7ix—{qmx — qpnx)—-na—qmx 
-\-pqnx={n—qm-\-pqn)x  ;  (because  qc=.qmx — qpnx)  therefore 
i»  measures  d  by  the  units  in  n  —  qm-\-pqn^  and  as  it  also 
measures  a,  and  6,  the  numbers,  or  quantities  a,  b,  and  d  have 
a  common  measure.  Now  the  greatest  common  measure  of  d 
is  itself ;  consequently  d  is  the  greatest  common  measure  of 
a  and  b. 

133.  To  find  the  greatest  common  measure  of  three  num- 
bers, or  quantities,  «,  ^,  c  ;  let  d  be  the  greatest  common 
measure  of  a  and  Z>,  and  x  the  greatest  common  measure  of  d 
and  c  ;  then  x  is  the  greatest  common  measure  of  a,  5,  and  c. 
For,  as  <z,  Z>,  and  d  have  a  common  measure  ;  if  d  and  c  have 
also  a  common  measure,  that  same  number  or  q^Jhtity  will 
measure  a,  ^,  and  c  ;  and  if  x  be  the  greatest  common  measure 
of  d  and  c,  it  will  also  be  the  greatest  common  measure  of  a, 
b,  and  c. 

And,  in  like  manner,  if  there  be  any  number  of  quantities  ; 
<z,  5,  c,  d,  &c. ;  and  that  x  is  the  greatest  common  measure 
of  a  and  b  ;  y  the  greatest  common  measure  of  x  and  c  ;  z  the 
greatest  common  measure  of  y  and  d ;  &c.  &c.  ;  then  will  y 
be  the  gTeatest  common  measure  of  «,  b,  and  c ;  2^  the  great- 
est common  measure  of  a,  5,  c,  and  <^ ;   &c.  &c. 

134.  The  preceding  method  of  demonstration  is  similar  to 
that  given  by  Bridge  in  his  Treatise  on  the  Elements  of  Alge- 
bra.  The  following  is  according  to  the  manner  of  Garnier. 
Thus,  to  find  the  greatest  common  divisor  of  any  number  of 
quantities  A,  w,  C,  &;c.,  it  is  sufficient  to  know  the  method  of 
finding  the  greatest  common  divisor  of  two  numbers  or  quan- 
tities. For  this  purpose,  we  will  at  first  seek  the  greatest  com- 
mon divisor  D  of  the  quantities  A  and  B,  then  the  greatest 
common  divisor  D'  of  D  and  C,  and  so  on,  and  finally  the  last 
greatest  common  divisor  will  be  that  which  was  required. 

Let,  in  order  to  demonstrate  it,  the  three  quantities  be  A, 
B,  C  ;  we  will  have 

7 


62  ALGEBRAIC  FRACTIONS. 

m  and  n  are  necessarily  prime  to  one  another,  otherwise  D 
would  not  be  the  greatest  common  divisor  of  A  and  B  ;  r  and 
q  are  also  prime  to  one  another,  in  order  that  D^  may  be  the 
greatest  common  divisor  of  D  and  C.  Now  rD^the  greatest 
common  divisor  of  A  and  B,  cannot  be  the  greatest  common 
divisor  of  A,  B,  and  C,  unless  that  r  be  equal  to  q,  or  a  factor 
of  q  ;  but  r  and  q  being  prime  to  one  another ;  D''  remains  the 
greatest  common  divisor  of  A,  B,  and  C. 

135.  As  the  problem  of  finding  the  greatest  common  divisor 
of  any  two  quantities  A  and  B^  is  the  same  as   to  reduce  a 

fraction  ^^  to  its  most  simple  expression ;  because  that  in  di- 

viding  A  and  B  by  their  greatest  common  divisor,  we  have 
the  two  least  quotients  possible ;  admitting  this  enunciation, 
and  supposing  A>B. 

The  greatest  common  divisor  of  A  and  B,  cannot  exceed 
B  ;  it  could  be  B  itself,  which  we  can  readily  know,  if  we 
perforni^e  division  of  A  by  B,  which  gives 

^=?+g- .  .  . .  (1), 

q  being  the  integral  quotient,  and  R  the  remainder,  if  A  is  not 

exactly  divisible  by  B.     The  fraction  ^  being  changed  into  q 

Tf  R  R 

4- —  cannot  be  reduced  unless  that  .pr-  or  its  reciprocal  -pj-  is 
B  •    B  '^  R 

reducible,  because  q  is  an  integral  quantity  which  is  always 

irreducible ;  or  B  being  >  R,  the  quantity  which  ought  to  re- 

duce  r— ,  cannot  exceed  R,  it  might  be  R  itself,  which  we  will 

R 
know  in  performing  the  division  of  B  by  R,  which  gives 

q^  being  the  integral  part  of  the  quotient,  and  R'  the  remain- 
der <R  ;  we  say  still  that  the  reduction  of  rr-  depends  on  that 

R' 

of  — ,  or  its  reciprocal,  because  that  q^  is  an  irreducible  quan- 

R 
tity  ;  so  that  by  continuing  in  this  manner  we  shall  have  the 
following  decomposjtioi^s  : 


ALGEBRAIC  FRACTIONS.  63 

^=q  +^  .  .  .  .  (3), 

We  see  very  clearly  that  the  quantity  which  ought  to  reduce 

A  R         R 

r^-  is  that  which  must  reduce  ^  or  ^,   which   must    reduce 

=j-  or  :pr7,  which  must  reduce  ^rr  or  r-— , 
K       li  K,        n. 

If,  for  example,  R'"=0,  this  quantity  cannot   be    greater 

than  R"  ;  R"  is  therefore  the  greatest  quantity  which  can  re- 

A 
duce  the  fraction  ^  ;  consequently  it  is  the  greatest  common 

divisor  of  A  and  B. 

136.  Let  R''=iO  and  R'^^i  :  unity  will  be^  according  to 
what  has  been  above  demonstrated,  the  greatest  common  di- 

visor  of  A  and  B  ;  the  fraction  =5-  will  therefore  itself  be  the 

JD 

most  simple  expression,  that  is,  it  will  be  irreducible.  Re- 
ciprocally,  the  last  divisor  being  unity ^  we  may  conclude  that  the 
fraction  proposed,  is  irreducible,  or  in  its  lowest  terms. 

137.  It  may  also  be  shown,  that  the  greatest  common  mea- 
sure of  two  quantities  will,  in  no  respect,  be  altered,  by  mul- 
tiplying or  dividing  either  of  them  by  any  quantity  which  is 
not  a  divisor  of  the  other,  or  that  contains  no  factor  which  is 
common  to  both  of  them  ;  thus,  let  the  quantities  ab  and  ac 
be  taken,  of  which  the  common  measure  is  a ;  then,  if  ab  be 
multiplied  by  d,  they  will  become  abd,  and  ac ;  where  it  is 
evident  that  a  is  the  common  measure,  as  before.  And,  con- 
versely, if  the  first  of  the  two  quantities  abd,  ac,  be  divided 
by  d,  they  will  become  ab,  ac,  where  a  is  still  the  common 
measure. 

138.  But  it  will  not  be  the  same  if  one  or  two  of  the  quan- 
tities be  multiplied  or  divided  by  a  quantity  which  is  a  divisor 
of  the  other,  or  has  a  common  factor  with  it ;  for  if  the  first 
of  the  two  quantities  ab,  ac,  be  multiplied  by  c,  they  will  be- 
come abc,  ac,  of  which  the  common  divisor  is  ac,  instead  of 
u  ;  and,  conversely,  if  the  first  of  the  two  quantities  abc  and 
ac,  be  divided  by  c,  they  will  become  ab  and  ac  ;  of  which 
the  common  divisor  is  a,  instead  of  ac. 

139.  Hence,  if  the  numbers  or  quantities  be  mncl^,  pqcN' ; 
the  common  factor  c,  to  simplify  the  operation,  may  be  sup- 
pressed, observing,  in  the  meantime,  after  having  found  the 


64  ALGEBRAIC  FRACTIONS. 

greatest  common  divisor  a,  of  the  two  quotients  N  and  N',  to 
multiply  it  by  this  factor  c,  and  the  product  will  be  the  great- 
est common  divisor  sought.  Also,  if  a  factor  d  is  introduced 
into  the  two  quantities,  it  is  necessary  to  divide  the  greatest 
common  divisor  by  this  factor. 

140.  As  the  foregoing  demonstration  may  be  extended  to 
any  algebraic  quantities  whatever,  we  are  therefore  conducted 
to  this  practical  rule. 

To  find  the  greatest  common  divisor  of  two  or  more  compound 
algebraic  quantities, 

RULE. 

141.  Arrange  the  two  quantities  according  to  the  order  of 
their  powers,  ^nd  divide  that  which  is  of  the  highest  dimen- 
sions by  the  other,  having  first  expunged  any  factor  that  may 
be  contained  in  all  the  terms  of  the  divisor  without  being 
common  to  those  of  the  dividend  ;  then  divide  this  divisor  by 
the  remainder,  simplified,  if  necessary,  as  before  ;  and  so  on, 
for  each  remainder  and  its  preceding  divisor,  till  nothing  re- 
mains :  then  the  divisor  last  used  will  be  the  greatest  com- 
mon divisor  required.  And  the  greatest  common  divisor,  of 
more  than  two  compound  quantities,  is  found  in  like  manner  ; 
by  finding  in  the  first  place  the  greatest  common  divisor  of 
two  of  them,  as  above,  and  then  of  that  common  divisor  and 
the  third,  and  so  on.  The  last  divisor,  thus  found,  will  be  the 
greatest  common  divisor  of  all  the  quantities. 

Example  1.  The  greatest  common  divisor  of  the  compound 
quantities  3a^  — Sa^i-f-aZ^^  — Z>3  and  Aa^h—^aly^-^-h^,  is  required. 
Dividend.  Divisor. 


3a3-3a2Z»+  ah'^—h^ 
4 

12a3  — 12a25  +  4a&2— 4&3 
12a3-15a2&-|-3a62 


(4a25  — 5a62_^Z>3)^5_, 
4a2  _5a6  +&2 


Partial  quot.  3a 


(3a2Z>+   a52_4Z^3)-ri  = 
3a2  +   ah  —452 
4 


12a24-  4a5— 1662 
12a2— 15a6+   362 


19a6— 1962 


Divisor. 
4a'^—5abi-b^ 

Partial  quot.  3. 


ALGEBRAIC   FRACTIONS.  65 

Dividend.  Divisor. 


4a2— 5ai+i2 
4a^ — Aab 


ab  +  b^ 


(I9ab-i9b^)'r-l9b: 
a — b 


Quot.  4a — b 


Here  the  quantities  are  already  arranged  according  to  the 
powers  of  the  letter  a  :  the  first  is  taken  for  a  dividend,  and 
the  second  for  a  divisor.  In  the  first  place,  the  factor  b  is 
found  in  every  term  of  the  divisor,  and  not  in  every  term  of 
the  dividend ;  therefore,  the  divisor  is  divided  by  the  factor  b, 
and  the  result  is  4a'^  —  5ab-\-b'^ ;  but  the  first  term  of  this  re- 
sult will  not  divide  exactly  that  of  the  dividend,  on  account  of 
the  factor  4,  which  is  not  in  the  dividend  ;  the  dividend  is 
therefore  multiplied  by  4  in  order  to  render  the  division  of  their 
first  terms  complete.  Now,  the  dividend  I2a^ — 12 a'^b-\- 4 ab"^  — 
4b^  is  divided  by  the  divisor  4u?—bab-\-b'^,  and  the  partial  quo- 
tient is  3a.  Multiplying  the  divisor  by  this  quotient,  and  sub- 
tracting the  product  from  the  dividend,  the  remainder  is  3a^b 
-|-a62_4^3^  a,  quantity  which,  according  to  (Art.  135),  must 
still  have  with  '4a^ — bab-^-b"^  the  same  greatest  common  divisor 
as  the  first. 

Suppressing  the  factor  b,  common  to  all  the  terms  of  the 
remainder,  or,  which  is  the  same,  dividing  the  remainder  by  5, 
and  multiplying  the  result  by  4,  to  render  possible  the  division 
of  its  first  term  by  that  of  the  divisor,  we  have  then  for  the 
dividend  the  quantity 

12a24.4ai— 1662, 

and  for  the  divisor  the  quantity 

4o2_5aJ-h62; 
the  partial  quotient  is  3. 

Multiplying  the  divisor  by  the  quotient,  and  subtracting  the 
product  from  the  dividend,  the  remainder  is 

19a6  — 1962, 
and  the  question  is  now  reduced  to  finding  the  greatest  common 
divisor  of  19a6— 19^2  and  4a'^  —  bab-\-b'^. 

But  the  letter  a,  according  to  which  the  division  has  been 
performed,  being  of  the  second  degree  in  the  divisor,  and  only 
of  the  first  in  the  remainder  ;  it  is  necessary  therefore  to  take 
the  last  divisor  for  a  new  dividend,  and  the  remainder  for  a  new 
divisor. 

Having,  at  the  commencement  of  this  new  division,  divided 
the  divisor  I9ab  —  I9b'^  by  the  factor  196,  common  to  all  its 
7* 


66 


ALGEBRAIC   FRACTIONS. 


terms,  and  which  is  not  at  all  common  to  those  of  the  dividend  : 
therefore  the  dividend  is  Aa^— 5 ab-\rb^,  the  divisor  a—b,  and 
the  quotient  4a— b  ; 

The  operation  is  completed,  because  nothing  remains  ;  and 
consequently,  (Art.  135),  a—b  is  the  greatest  common  divisor 
sought. 

If  we  divide  the  two  proposed  quantities  by  a— ft,  the  quo- 
tients will  be 

3a2+i2  and  4ab—b^  : 
Whence,  the  two  given  quantities  are  thus  decomposed  as 
follows  : 

(3a2+62)x(a-&),  (4ab-b^)x(a'-b). 
Ex.  2.  Required  the  greatest  common  divisor  of  30^— .2a— 1 
and4a3— 2a2_3a+l. 

Dividend.  Divisor. 

4a3_2a2_3^_|_l     3«2_2a_l 

3 


12a3— 6a2_9a+3 
12a3— 8a2_4a 


2a2. 
3 


5a+3 


6^2— I5a+g 
6a2_  4a— 2 


Partial  quot.  4a 

Divisor. 
3a2_2a-l 


Partial  quot.  2 


(  — lla+ll)-4-— 11 
Dividend. 

a-l 


3«2- 

-2a- 

-1 

3a2- 

-3a 

a- 

-1 

a- 

-1 

Complete  quot.  3a  4"  1 


In  the  above  operation,  the  remainder  —  lla-f-H  is  divid- 
ed by  —11,  (its  greatest  simple  divisor  with  a  negative  sign), 
so  as  to  make  the  leading  term  positive  :  or,  which  is  the  same, 
if  any  of  the  divisors,  in  the  course  of  the  operation,  become 
negative,  they  may  have  their  signs  changed,  or  be  taken 
affirmatively,  without  altering  the  truth  of  the  result ;  thus,  in 
the  above  operation,  changing  the  signs  of  —lla+ 11,  it  be- 
comes 11a— 11,  and  dividing  11a— 11  by  its  greatest  simple 
divisor  11,  we  have  a— 1,  as  before. 


ALGEBRAIC  FRACTIONS. 


67 


Therefore  a— I  is  the  greatest  common  divisor  sought, 
and  the  two  ajiven  quantities  may  be  readily  decomposed, 
thus; 

(3a+l)x(a-l),  (4a2+2a-l)x(a— I). 

Ex.  3.  Required  the  greatest  common  divisor  of  a^—P, 
a^-{-2a^+2ab^+P,  and  a'^-\-a^b^-\-b\ 

In  the  first  place,  the  greatest  common  divisor  of  a^—b^ 
and  a^+2aH-{-2ab^-\-b^,  is  a^-\-ab-{-b^,  which  is  found  thus  ; 
Dividend.  Divisor. 


a^+2a'^b+2ab^-\-b^ 
a3  -63 


a3-63 


Partial  quot.  1 


(2a'^b-{-2ab^+2b^)-^2b=z 
Dividend, 
a^-b^  a'^+ab-hb^ 

a^J^a^b-\-ab^ 


-a^b-ab^-b^ 
.a^—ab^-b^ 


Complete  quot.  a — b 


Hence,  the  greatest  common  divisor  of  a^—¥  and  a^-\-2a'^b 
-{-2ab^-\-b^,  is  a^-{-ab-{-b^  ;  and  the  greatest  common  divi- 
sor of  a'^-{-ab-\-b^  and  a^+a^^^-f  M,  is  found  to  be  a'^—ab-\-b^, 
thus; 

Dividend. 


—a^  +b* 

— a^b  — a^^ — aP 

aW-^ab^^-b^ 
fl2fe2_^a63+M 


Divisor. 
a^-\-ab-{-b'^ 


Quotient, 
a^—ab+b"^ 


Consequently  a^-\-ab-\-b'^  is  the  greatest  common  divisor 
which  was  required  ;  and  dividing  each  of  the  given  quanti- 
ties by  this  divisor,  we  will  thus  decompose  them  as  follows : 
(a-6)  [a^J^ab-irb'^),  (a+*)  (a^-^ab-\-b'^),  (a^^ab-^b^)  (a^-\- 
ab+b^y 

142.  It  has  been  remarked  (Art.  136),  that  if  the  last  divi- 
sor be  unity,  and  the  remainder  nothing ;  then  the  fraction  is 


68 


ALGEBRAIC  FRACTIONS. 


already  in  its  lowest  terms  ;  this  observation  is  applicable«to 
numbers,  and  as  in  algebraic  quantities,  the  greatest  simple 
divisor  may  be  readily  found  by  inspection. 

Now,  it  only  remains  to  discover,  if  compound  algebraic 
quantities  can  admit  of  a  compound  divisor. 

If,  by  proceeding  according  to  the  Rule  (Art.  141),  no 
compound  divisor  can  be  found,  that  is,  if  the  last  remainder 
be  only  a  simple  quantity ;  we  may  conclude  the  case  pro- 
posed does  not  admit  of  any,  but  is  already  in  its  lowest  terms. 

Ex.  4.  Required  the  greatest  common  divisor  of  a^-j-ax-^- 
sc^  and  a^-\-2a'^x-\-3ax'^-\-4x^.  It  is  plain  by  inspection  that 
they  do  not  admit  of  any  simple  divisor ;  then  the  operation 
according  to  the  rule  will  stand  thus ; 

Dividend.  Divisor. 


a^-\-2a'^x+3ax^+Ax^ 

a^x~\-2ax'^-\-4x^ 
a^aj-j-   ax^-\-  x^ 


a^-{-ax-^x'^ 


Dividend. 
a^-\-   aa?+   x"^ 
a^-{-3ax 

— 2ax-{-  x^ 
— 2ax — 6a?2 


Partial  quot.  a-{-x 

[ax'^-\-3x^)-^x^  = 


a-\-3x 


Partial  quot.  a — 2x 


*  +7x^ 

Here,  the  last  remainder  is  found  to  be  the  simple  quantity 
7x^  ;  we  may  therefore  conckide  that  the  given  quantities  do 
not  admit  of  any  divisor  whatever. 

143.  When  the  quantity  which  is  taken  for  the  divisor  con- 
tains many  terms  where  the  letter,  according  to  which  we 
have  arranged,  has  the  same  exponent ;  then  every  succes- 
sive remainder  becomes  more  complicated  than  the  preceding 
one  ;  in  this  case,  Analysts  make  use  of  various  artifices, 
which  can  only  be  learned  by  experience. 

Ex.  5.  Required  the  greatest  common  (Jivisor  of  a^J+oc^ 
— cZ^  anda6— ac+(?2. 

Dividend.  Divisor. 


a^J-j-ac^ — d^ 
a'^b—a^c-\-ad'^ 

rem.     a'^c-\-ac'^—ad^—d^ 


ah—ac-\-d^ 


Partial  quot.  a 


Dividing  at  first  a%  by  ah,  we  find  for  the  quotient,  a ; 


ALGEBRAIC  FRACTIONS. 


69 


multiplying  the  divisor  by  this  quotient,  and  subtracting  the 
product  from  the  dividend,  the  remainder  contains  a  new  term, 
a^c,  arising  from  the  product  of  —ac  by  a. 

By  proceeding  after  this  manner  there  will  be  no  progress 
made  in  the  operation  ;  for,  taking  cP-c-\-ac^—ad?'^d^  for  a 
dividend,  and  multiplying  it  by  6,  to  render  possibm  the  divi- 
sor by  a6,  we  will  have 

Dividend.  Divisor. 

a?bc-\-abc'^  —  abd? — hd^  ab — ac-\-d? 

d^bc—a^c'^-\-acd^ 


rem. 


'^c^^abc^—acd?—abd?  —  bd'^ 


Partial  quot. 


and  the  term  —ac  will  still  r.eproduce  a  term  aV,  in  which  the 
exponent  of  a  is  2. 

To  avoid  this  inconveniency,  we  must  observe  that  the  di- 
visor ab  —  ac-\-d?z:za{b—c)-\-d?,  reuniting  the  terms  ab — ac 
into  one,  and  putting,  to  abridge  the  calculations,  b  —  c=zm; 
we  will  have  for  the  divisor  a7n-{-d^  ;  it  is  necessary  to  mul- 
tiply all  the  dividend  a'^b-\-ac^—d^  by  the  factor  m,  for  the  pur- 
pose of  finding  a  new  dividend  whose  first  term  would  be  divi- 
sible by  the  quantity  am  forming  the  first  terra  of  the  divisor ; 
the  operation  will  become, 

Dividend.  Divisor. 

a^bm-^-ac^m—d^m         am-{-d^ 

a^bm-{-abd^ 


1st  rem.    -^ac^m—abd^—d^m 
-\-ac^m-\-c^d^ 


Partial  quot. 
ab  +  c'^ 


2d  rem.  ~abd^ — c'^d^ — d^m 
By  the  first  operation,  the  terms  involving  a^  are  taken  away 
from  the  dividend,  and  there  remain  no  terms  involving  a  ex- 
cept in  the  first  power.  In  order  to  make  them  disappear,  we 
will  at  first  divide  the  term  ac^m  by  am,  and  it  gives  for  the 
quotient  c'^ ;  multiplying  the  divisor  by  the  quotient,  and  sub- 
tracting the  product  from  the  dividend,  we  will  have  the  second 
remainder  ;  taking  this  second  remainder  for  a  new  dividend, 
and  cancelling  in  it  the  factor  d^,  which  is  not  a  factor  of  the 
divisor,  it  will  become 

— ab — c^ — dm  ; 
multiplying  by  m,  we  shall  have 

Dividend.  Divisor. 


— abm- 
— abm- 


■c^m- 

■bd:^ 


■drr? 


rem.  -{-bd^—c^m—dm^ 


am-{-d^ 

Partial  quot.  — b. 


70  ALGEBRAIC  FRACTIONS. 

The  remainder,  hd?  —  c-m—dni^,  of  this  last  division  does  not 
contain  the  letter  a  ;  it  follows,  then,  that  if  there  exist  between 
the  proposed  quantities  a  common  divisor,  it  must  be  indepen- 
dent of  the  letter  a. 

Havin^arrived  at  this  point,  we  cannot  continue  the  divi- 
sion with<|fcspect  to  the  letter  a  ;  but  observing  that  if  there  be 
a  common  divisor,  independent  of  a,  of  the  two  quantities 
hd^  —  c^m  —  dm^  and  am-^-d^,  it  may  divide  separately  the  two 
parts  am  and  d^  of  the  divisor  ;  for,  in  general,  if  a  quantity  be 
arranged  according  to  the  powers  of  the  letter  a,  every  term 
of  this  quantity,  independent  of  a,  must  divide  separately  the 
quantities  by  which  the  different  powers  of  this  letter  are 
multiplied. 

In  order  to  be  convinced  of  wFiat  has  just  been  said,  it  is 
sufficient  to  observe,  that  in  this  case  each  of  the  proposed 
quantities  should  be  the  product  of  a  quantity  dependent  on 
G,  and  of  a  common  divisor  which  does  not  at  all  depend  on  it. 
Now,  if  we  have,  for  example,  the  expression 

A^^+BflS  +  Ca^+Da+E, 
in  which  the  letters  A,  B,  C,  D,  E,  designate  any  quantities 
whatever,  independent  of  a,  and  if  we  multiply  it  by  a  quantity 
M,  also  independent  of  «,  the  product, 

MAa4+MBa3  +  MCa2+MDa+ME, 
arranged  according  to  a,  will  still  contain  the  same  powers  of 
a  as  before  ;  but  the  coefficient  of  each  of  these  powers  will  be 
a  multiple  of  M. 

This  being  admitted,  if  we  substitute  for  m  the  quantity 
(i— c),  which  this  letter  represents,  we  shall  have  the  quan- 
tities 

hd'^^c\h-c)-c{h-c)\ 
-  a(5-c)+c?2; 
now  it  is  plain  that  h  —  c  and  d^  have  no  common  factor  what- 
ever :  therefore  the  two  pro^^osed  quantities  have  not  a  com- 
mon divisor. 

144.  The  greatest  common  divisor  of  two  quantities  may 
sometimes  be  obtained  without  having  recourse  to  the  general 
Rule.  Some  of  the  methods  that  are  used  by  Analysts  for  this 
purpose,  will  be  exemplified  by  the  following  Examples. 

Ex.  6.  Required  the  greatest  common  divisor  of  a^i^-f-^'^^^ 
-f  i*c2— aV— a3Z,c2— ^2^4,  and  d^b  +  ab'^-^h^—a''-c—abc-'hH. 

After  having  arranged  tKese  quantities  according  to  the 
powers  of  the  letter  a,  we  shall  have 

(52_c2)a4_|_(^,3_^,c2)a3-f-J4c2_52c4 

(5-c)a2-f(62_Z»c)a+63-J2c; 


ALGEBRAIC  FRACTIONS.  71 

it  may  at  first  be  observed,  tbat  if  they  admit  of  a  common  di- 
visor, which  should  be  independent  of  the  letter  a,  it  must  di- 
vide separately  each  of  the  quantities  by  which  the  different 
powers  of  a  are  multiplied,  (Art.  143),  as  well  as  the  quanti- 
ties h'^c^—b'^c^  and  P  —  b^c,  which  comprehend  not  at  all  this 
letter. 

The  question  is  therefore  reduced  to  finding  the  common 
divisors  of  the  quantities  b^ — c^  and  b — c,  and,  to  verify  af- 
terward, if,  among  these  divisors,  there  be  found  some  thai 
would  also  divide  b^—bc^  and  b'^—bc,  b*c^—b^c^  and  b^~b^c. 

Dividing  b'^ — c^  by  b — c,  we  find  an  exact  quotient  b-\-c  : 
b—c  is  therefore  a  common  divisor  of  the  quantities  b^—c^ 
and  b  —  c,  and  it  appears  that  they  cannot  have  any  other  di- 
visor, because  the  quantity  b—c  is  divisible  but  by  itself  and 
unity.  We  must  therefore  try  if  it  would  divide  the  other 
quantities  referred  to  above,  or,  which  is  equally  as  well,  if 
it  would  divide  the  two  proposed  quantities  ;  but  it  will  be 
found  to  succeed,  the  quotients  coming  out  exactly, 
(b+c)a^-{-{b^-^bc)a^'frb^c^+b^c^; 
and         a'^-\-ba-\-b^. 

In  order  to  bring  these  last  expressions  to  the  greatest  pos- 
sible degree  of  simplicity,  it  is  expedient  to  try  if  the  first  be 
not  divisibfe  by  b-{-c  ;  this  division  being  effected,  it  succeeds, 
and  we  have  now  only  to  seek  the  greatest  common  divisor  of 
these  very  simple  quantities  ; 

a^-^baHb^C^,  and  a?-^ba+b^. 

Operating  on  these,  according  to  the  Rule,- (Art.  141),  we 
will  arrive,  after  the  second  division,  at  a  remainder  contain- 
ing the  letter  a  in  the  first  power  only  ;  and  as  this  remainder 
is  not  the  common  divisor,  hence  we  may  conclude  that  the 
letter  a  does  not  make  a  part  of  the  common  divisor  sought, 
which  is  consequently  composed  but  of  the  factor  b — c. 

Ex.  7.  Required  the  greatest  common  divisor  of  (d^ — c^) 
Xa^+c*  — fZV  and  4:da:^—(2c'^+4cd)a-i-2c^. 
Arranging  these  quantities  according  to  d,  we  have 

(a2_c2)£Z2  +  c4-a2c2,  or  (a2_c2)(^2_(^2_c2)c2, 

and         {4a^—4ac)xd—(a—c)x2c^; 

it  is  evident,  by  inspection  only,  that  a^ — c'^  is  a  divisor  of  the 
first,  and  a — c  of  the  second.  But  d^  —  c^  is  divisible  by  a — c  ; 
therefore  a—c  is  a  divisor  of  the  two  proposed  quantities  :  Di- 
viding both  the  one  and  the  other  by  a  —  c,  the  quotients  will 
be 

(a+c)x((^— c2),  and  4a(?— 2c2; 


72  ALGEBRAIC  FRACTIONS. 

which,  by  inspection,  are  found  to  have  no  common  divisor 
consequently  a—c  is  the  greatest  common  divisor  of  the  pro- 
posed quantities.  , 

Ex.  8.  Required  the  greatest  common  divisor  of  y^—x^  and 
y^ — y^x— yoj^-j-o?^.  Ans.  y^ — ^,2^ 

Ex.  9.  Required  the  greatest  common  divisor  of  a*—b^  and 
a^—¥.  Ans.  a^—lp-. 

Ex.  10.  Required  the  greatest  common  divisor  of  o^+a^^ — 
ah'^—h'^  and  a'^-^aW^-h'^.  Ans.  a^-^ab-^-y^. 

Ex.  11.  Required  the  greatest  common  divisor  of  a?-— 'lax 
■\-x^  and  c? — oP'X — ax^-\'x'^.  Ans.  a^ — 2ax-\-x^. 

Ex.  12.  Find  the  greatest  common  divisor  of  6a;^ — %yx^-\- 
2y2a;  and  12a;2 — 15ya;+3y2.  Ans.  x — y. 

Ex.  13.  Find  the  greatest  common  divisor  of  3662a6_  18^2^5 
— 27 J2a4-}- 952^3  and  27^2^5  _i  852^4  _  952^3. 

Ans.  9^»2a*— 9i2a3, 

Ex.  14.  Find  the  greatest  common  divisor  of  (c—</)a2^ 
{2hc--Ud)a-\-{hH-hH)  and  {hc—hd^c^—cd)a^{pd-\-hc^-' 
b'^c — bed).  Ans.  c—d, 

Ex.  15.  Find  the  greatest  common  divisor  of  x^-\-Qx'^-{- 
27a;— 98  and  a;2+12a;— 28.  Ans.  x—2. 

§  III.    METHOD    OF  FINDING  THE  LEAST  COMMON  IJIULTIPLE  OF 
TWO  OR  MORE   QUANTITIES. 

145.  The  least  common  multiple  of  two  or  more  quantities 
is  the  least  quantity  in  which  each  of  them  is  contained  with- 
out a  remainder.  Thus,  20abc  is  the  least  common  multiple 
of  ba,  AaCj  and  2b. 

146.  The  least  common  multiple  of  any  number  of  quanti- 
ties, literal  or  numeral,  monomiS  or  polynomial,  may  be  easily 
found  thus  : 

Resolve  each  quantity  into  its  simplest  factors^  putting 
the  product  of  equal  factors  when  there  are  any  in  the  form  of 
powers,  then  multiply  all  together  the  highest  powers  of  every 
root  concerned,  and  the  product  vnll  be  the  least  pgrnmon  multi- 
ple required. 

Ex.  1.  Required  the  least  common  multiple  of  aWx,  acbx^^ 
abc'^d. 

Here  the  quantities  are  already  exhibited  in  the  form  re- 
quired.    Therefore  the  least  common  multiple  is  a^'^c^dx^. 

Ex.  2.  Required  the  least  common  multiple  of  2cP'Xy  \ax^y 
and  ^x^. 


ALGEBRAIC  FRACTIONS.  73 

Here  the  literal  quantities  are  already  in  the  form  requir- 
ed. The  coefficients  resolved  into  their  simplest  factors  be- 
come 2,  22,  2x3.     The  least  common  multiple  is  therefore 

Ex.  3.  Required  the  least  common  multiple  of  l2a^i/(a-\-b)j 
6ay+12a2  by^fdabY,  and  Aa^. 

These  quantities  resolved  into  their  simplest  factors  become 

2^x3xa^y(a  +  b) 

2  x3Xaf{a+bf 

22  X  a2y2 
Hence  the  least  common  multiple  required  is  22  x  3  X  a^y^ 

{a-hbY=i2aY{a-\-f>f- 

Ex.  3.  Required  the  least  common  multiple  of  8a,  4a2,  and 
I2ab.  Ans.  24a2^. 

Ex.  4.  Required  the  least  common  multiple  of  a2— &2^  a+i, 
and  a^-\-b^.  Ans.  a'^  —  b*. 

Ex.  5.  Required  the  least  common  multiple  of  72a,  1 5b, 
9ab,  and  3a2.  Ans.  135a2^>. 

Ex.  6.  Required  the  least  common  multiple  of  a^+Sa^b-i- 
3ab^+b\  a^-j-2ab  +  b^,  a^—b^.  Ans.  a^-^2a^—2aP-^b^. 

Ex.  ^.  Required  the  least  common  multiple  of  a-\-bj  a—b, 
a^-^-ab-^-b^,  and  a^  —  ab-^b^.  Ans.  a^—¥ 


^  IV.    REDUCTION    OF    ALGEBRAIC    FRACTIONS. 
CASE    I. 

To  reduce  a  mixed  quantity  to  an  improper  fraction. 

RULE. 

147.  Multiply  the  integral  part  by  the  denominator  of  the 
fraction,  and  to  the  product  annex  the  numerator  with  its  pro- 
per sign :  under  this  sum  place  the  former  denominator,  and 
the  result  is  the  improper  fraction  required. 

Ex.  1.  Reduce  3x  +^  to  an  improper  fraction. 

The  integral  part  3x,  multiplied  by  the  denominator  5a  of 
the  fraction  plus  the  numerator  (2b),  is  equal  to  3a:X5a-|-26 
=  l5ax+2b', 

Hence, is  the  fraction  required. 

3a? 
Ex.  2.  Reduce  5a to  an  improper  fraction. 

8 


74  ALGEBRAIC   FRACTIONS. 

Here  5axy=^^ay\  to  this  add  the  numerator  with  its  pro- 
per sign,  viz.  —3a; ;  and  we  shall  have  5ay— 3a;. 

Hence,  — =- is  the  fraction  required. 

y 

q2 yZ 

Ex.  3.  Reduce  x^ —  to  an  improper  fraction. 

Here,  x^Xx^zx^  \  adding  the  numerator  a^ — y^  ^j^A  its  pro- 
per sign :  It  is  to  be  recollected  that  the  sign  •—  affixed  to  the 

q2 J.2 

fraction —  means  that  the  whole  of  that  fraction  is  to  be 

X 

subtracted,  and  consequently  that  the  sign  of  each  term  of  the 
numerator  must  be   changed,  when  it  is  combined  with  x^ 

hence  the  improper  fraction  required  is ~.     Or,  as 

^2 «f2  q2  _1_  «»2         |f2  .^ ^2 

^—= :i-  — -^ ;  (Art.  67),  the  proposed  mixed 

X  X  x  ^ 

(jp- j/2  |.2 ^2 

quantity  a;^ ^,  may  be  put  under  the  from  x^-\-- » 

X  X 

which  is  reduced  as  Ex.  1.  Thus,x'^Xx-{-i/^—a^=x^-\h/^—a^'f 
,             \  ,  y^—a^     a;3+y2_Q2 
hence,  x^+- = . 

X  X 

3/p2 a4-7 

Ex.  4.  Reduce  5a^-\ to  an  improper  fraction. 

liax 

Here,  5a'^x2axz=z\0a'^x  ;  adding  the  numerator  3a?2— a+7 

to  this,  and  we  have  10a%-|-3a;^— a+7. 

__            lOA-f  3a;2-a+7  .,,.,.  .     , 

Hence, is  the  fraction  required. 

2ax 

Ex.  5.  Reduce  Ax"^ to  an  improper  fraction. 

Here,  Ax'^x2ac=Sacx'^,m.  adding  the  numerator  with  its 
proper  sign  ;  the  sign  — •  prenxed  to  the  fraction  — signi- 
fies that  it  is  to  be  taken  negatively,  or  that  the  whole  of  that 
fraction  is  to  be  subtracted  ;  and  consequently  that  the  sign 
of  each  term  of  the  numerator  must  be  changed  when  it  is 

combined  with  Qacx"^ ;  hence, is  the  fraction  re- 

2ac 

.  ,  ^  ^ab-{-c  .  — 3a5— c  —Zah—c  ,.^ 
quired.      Or,    as __= +^__  =_g_-     (Art. 

108) ;  hence  the  reason  of  changing  the  signs  of  the  numera- 
tor is  evident. 


ALGEBRAIC  FRACTIONS.  75 


Ex.  6.  Reduce  x to  an  improper  fraction. 

X 


Ans. 

X 

d?"  '\'  c  * 

Ex.  7.  Reduce  ab —  to  an  improper  fraction. 

Ox 

.        5abx — a^ — c 

Ans. . 

5x 

Ex.  8.  Reduce  ax"^ to  an  improper  fraction. 

a'^x^-Sb 

Ans. . 

a 

Ex.  9.   Reduce  a—x-\ to  an  improper  fraction. 

X 

.         a^—x^ 
Ans.  -. 

X 

4^ 9 

Ex.  10.  Reduce  3x^ —  to  an  improper  fraction. 

21aa;2_4a;4-9 

Ans. ■ — 

7a 

2x 5 

Ex    11.  Reduce  5x —  to  an  improper  fraction. 

o 

13i«;+5 
Ans.  -^— . 

Ex.  12.  Reduce  14-2a? — to  an  improper  fraction. 

DX 

a;4-10a!24.4 

Ans. , 

5a; 


CASE  II. 
To  reduce  an  improper  fraction  to  a  whole  or  mixed  quantity. 

RULE. 

148.  Observe  which  terms  of  the  numerator  are  divisible 
by  the  denominator  without  a  remainder,  the  quotient  will  give 
the  integral  part ;  and  put  the  remaining  terms  of  the  nume- 
rator, if  any,  over  the  denominator  for  the  fractional  part ; 
then  the  two  joined  together  with  the  proper  sign  between 
them,  will  give  the  mixed  quantity  required. 


76                   ALGEBRAIC  FRACTIONS. 
Ex.  1.  Reduce to  a  mixed  quantity. 

X 

T-.         x^-\-2ax'^  ^     •     1      •  1  ,    5    .      , 

Here, =:x-{-2a  is  the  integral  part,  and  —  is  the 

fractional  part ;     • 

therefore  x-\-2a-\ — -  is  the  mixed  quantity  required. 

X 


^8    I    jp4 J.4  J_  y8 

Ex.  2.  Reduce  -— — '-—- — <-  to  a  whole  quantity. 

<K4_j_a,2y2_|_y4 


Dividend.  Divisor. 

x^-\-x*y^+y^ 


■  a;^y2_|_y8 


—  a;6y  2  —  ^4y4 — ^2y  6 


3-4^4  _f_j,2y6_|_y8 


Quotient. 

a;4_^2y2_j_y4 


Here  the  operation   is  performed   according  to  the  rule 

(Art.  93),  and  the  quotient  x'^ — x^y^-^-y*"  is  the  whole  quantity 

required. 

-i-i      r^    T-.    T        <3!^ — 2b^  .      , 

Ex.  3.  Reduce to  a  mixed  quantity. 

X 

Here,  — =:a  is  the  integral,  and the   fractional   part ; 

therefore  a is  the  mixed  quantity  required. 

qq2 ^2    I    J 

Ex.  4.  Reduce -^  to  a  mixed  quantity. 

x-\-a 

x-^a)x^^a'^-{-b(x—a-\ -the  mixed  quantity  required. 

x+a 

x^-\-ax 

— ax — a^ 
—ax—a^ 

*  +* 
Here  the  remainder  b  is  placed  over  the  denominator  x-\-a, 

and  annexed  to  the  quotient  as  in  (Art.  89). 

Ex.  5.  Reduce — ^r-7 ■ to  a  mixed  quantity. 

3ab 


ALGEBRAIC  FRACTIONS.  77 

3a^b^-\-6ab        ,.«.,. 

Here —7 =ab+2  is  the  integral  part, 

odo 

-  — 2a;+2c         2a:— 2c       .  2c— 2a:  ,  ,       ,^ov    •      r     r 

and — :r-r- — = -— ,-=:-^ — — -—  (Art.  128),  is  the  frac- 

3ab  Sab        ^    Sab      ^  " 

tional  part ; 

,  ,  „     2a;— 2c          ,  .  „  .   2c— 2a:  .      ,  .      , 

.  • .  ab+2 —, — ,  or  aft +2  H —j—  is  the  mixed  quantity- 
required. 

2lax^ 4x-\-9 

Ex.  6.  Reduce to  a  mixed  quantity. 

Ans.  3a;2 — . 

.     7a 
_       ^    ^    .         8a:2y2-3aa:-6&  .      ^ 

Ex.  7.  Reduce  —     a  i to  a  mixed  quantity. 

<jj4 ^4 

Ex.  8.  Reduce  -ytt^  ^^  ^  whole  quantity. 

Ans.  aj2— a^. 
JiiX.  9.  Keduce — to  a  mixed  quantity. 

Ans.  3a-.l  +  _^— . 

■r^      in    T>   J         »*— 3a:2y2+4aa:  .      , 

Ex.  10.  Reduce  ^-^^r^ to  a  mixed  quantity. 

x^—Sy^  "^ 

Ans.  x^-{- 


-3y2 

„       , ,     ,,    ,        x^+Sax^—a^~b 

Ex.  1 1 .  Reduce — — to  a  mixed  quantity. 

x^-\-a^  ^  "" 

Sax^—b 
Ans.  a:^— a^-l 5-7— ^• 

^      ,«    r,   J        3a:2— 12aa;+y— 9a: 

Ex.  12.  Reduce — to  a  mixed  quantity. 

ijX 

y 

Ans.  a:— 4a— 3+  o^« 


8» 


76  ALGEBRAIC  FRACTIONS. 


CASE  III. 

To  reduce  a  fraction  to  its  lowest  terms,  or  most  simple 
expression. 

RULE. 

149.  Observe  what  quantity  will  divide  all  the  terms  both 
of  the  numerator  and  denominator  without  a  remainder :  Di- 
vide them  by  this  quantity,  and  the  fraction  is  reduced  to  its 
lowest  terms.  Or,  find  their  greatest  common  divisor,  accord- 
ing to  thfe  method  laid  down  in  (Art.  141) ;  by  which  divide 
both  the  numerator  and  denominator,  and  it  will  give  the  frac- 
tion required. 

Example  1. 

o    ,         I4x^-i-7ax^+28x       .     . 

Keduce  — — to  its  lowest  terms. 

21a;2 

The  coefficient  of  every  terra  of  the  numerator  and  deno- 
minator of  the  fraction  is  divisible  by  7,  and  the  letter  x  also 
enters  into  every  term  ;  therefore  7a?  will  divide  both  the  nu- 
merator and  denominator  without  a  remainder. 

_-       Mx^+7ax'^-\-28x     „  ,  ,         .  ^        ,21a;2     ^      , 
Now =:2x^-i-ax4-4,Rr\a—- — —3x:  hence 

7x  7x 

,       -       .       .     .^     ,  ^  .    2x'^^-ax-{-4: 

the  fraction  m  its  lowest  term  is . 

3a; 

T.      «    r.    J         SOa^b^c—dabc'^—Ua'^cH      .     , 

Ex.  2.  Keduce — to  its  lowest  terms. 

doaocx 

Here  the  quantity  which  divides  both  the  numerator  and 

denominator  without   a   remainder    is  evidently  6abc ;    then 

30a262c— 6a6c2— 12a2c2Z>  36abcx 

=:5ab — c—2ac  ;  and  -—— i —  =  oa; ; 

6abc  6abc 

^^  5ab — c—2bc  .     ,      ,,       .       •     •     , 

Hence is  the  fraction  in  its  lowest  terms. 

6a; 

a^  — &2 
Ex.  3.  Reduce  — — -,-  to  its  lowest  terms. 
«*— &4 

Here,  a*  -  b^  =  {a^  4  b^)  X  (a^  -  b%  (Art.  107.)  ;  and, 
consequently,  a^—b"^  will  divide  both  the  numerator  and  de- 

a2-62       , 
nominator  without  a  remainder  ;   that  is,  — — r^  =  1  =  new 


ALGEBRAIC  FRACTIONS.  79 

,(a2+&2)x(a2_52) 

numerator,  and  ^ r— 7:3 ■'=a--^b^=  new  denomma- 

a^ — tf^ 

tor ;  hence,    „  .  ,.,  is  the  fraction  in  its  lowest  ternnis. 

^       ,     T.    :.         a;4— 3aa;3— 8a2a;2^18a3a:  — 8a*       .     , 

Ex.  4.  Reduce — -— -— to  its  lowest 

terms. 

Here,  by  proceeding  according  to  the  method  of  (Art.  141), 
we  find  the  greatest  common  measure  of  the  numerator  and 
denominator  to  be  x'^-\-2ax — 2a2  ;  thus, 


x^—Sax^—Sa'^x'^-i-  I8a^x—8a^ 
ae* —  ax^ — 8a'^x'^-\-   6a^x 

—2ax^-{-l2a^x  —   Sa* 
—2ax^+  2a'^x^+l6a^x—12a^ 


x^  —  ax"^ — 8a^x-{-6a^ 
Partial  quot.  x — 2a 


remaind.  .  .  .  — 2a2a;2— 4u"V+4a*; 
1  —  2a2a;2_4Q;3^_|_4^4       _    ,    ^  ^  ^         . 

then, —-^ =x^  +  2ax  —  2a^  z=  the  next  di- 

— 2a^ 

visor  ; 

xi-\.2ax—2a^)x'^—  ax'^—8a'^x-\-Qa\x—'ia 
x'^'\-2ax'^ — 20^0? 


—  ^ax"^ —Qa"X-\-Qa^ 
— 3aa?2 — Ga^aj-j-Ga'^ 


And,  dividing  both  terms  by  the  greatest  common  measure, 
thus  found,  we  have  the  fraction  in  its  lowest  terms ;  but  the 
numerator,  divided  by  the  greatest  common  mep,sure,  gives  x 
—  3a,  as  above,  equal  to  the  new  numerator  ;  and  the  denomi- 
nator, divided  by  the  same,  gives  x^—5ax-{-4a'^ ;  thus, 


x^^Sax^-   8a'^x^+l8a^x-8a'*' 
a;*+2aa;3—  2a'2x^ 

— 5ax^ —  6a'^x'^-{-18a^x 
— 5aa;3 — 1  Oa^x'^  + 1  Oa^x 

4a^x"+   Sa^a;— 8a* 
4a2a;24-   Sa^aj—Sa* 


a;2+2aa;— 2a2 

Quotient. 
x'^—5ax-i-4a^ 


Hence,  the  fraction  in  its  lowest  terms  is 


80  ALGEBRAIC  FRACTIONS. 

x—3a 
x^ — 5ax^4a^' 
150.  In  addition  to  the  methods  pointed  out  in  (Art.  144), 
for  finding  the  greatest  common  divisor  of  two  algebraic  quan- 
tities, it  may  not  be  improper  to  take  notice  here  of  another 
method,  given  by  Simpson,  in  his  Algebra,  which  may  be  used 
to  great  advantage,  and  is  very  expeditious  in  reducing  frac- 
tions, which  become  laborious  by  ordinary  methods,  to  the 
lowest  expression  possible.  Thus,  fractions  that  have  in 
them  more  than  two  different  letters,  and  one  of  the  letters 
rises  only  to  a  single  dimension,  either  in  the  numerator  or  in 
the  denominator,  it  will  be  best  to  divide  the  numerator  or  de- 
nominator (whichever  it  is)  into  two  parts,  so  that  the  said 
letter  may  be  found  in  every  term  of  the  one  part,  and  be  to- 
tally excluded  out  of  the  other  :  this  being  done,  let  the 
greatest  common  divisor  of  these  two  parts  be  found,  which 
■will  evidently  be  a  divisor  to  the  whole,  and  by  which  the 
division  of  the  other  quantity  is  to  be  tried ;  as  in  the  follow- 
ing example. 

-n       ^    T^    1         x^4-ax^-\-bx'^—2a^x4-bax—2ba'^      .     , 

Ex.  5.  Reduce — ; — -— —-z toitslow- 

x^ — ox-^2ax — 2ao 

est  terms. 

Here  the  denominator  being  the  least  compounded,  and  b 
rising  therein  to  a  single  dimension  only  ;  I  divide  the  same 
into  the  parts  x^-{-2ax,  and  — bx — 2ab  ;  which,  by  inspec- 
tion, appear  to  be  equal  to  (a;+2a)a?,  and  {x-\-2a)x  —b. 
Therefore  x-\-2a  is  a  divisor  to  both  the  parts,  and  likewise 
to  the  whole,  expressed  by  (x-\-2a)x(x — b) ;  so  that  one  of 
these  two  factors,  if  the  fraction  given  can  be  reduced  to  lower 
terms,  must  also  measure  the  numerator :  but  the  former  is 
found  to   succeed,  the  quotient  coming  out  x^ — ax-\-bx—abi 

,           ,            1      r       .        •         1        1       x'^—ax-\-bx — ab 
exactly  :  whence  the  fraction  is  reduced  to ; , 

x—b 

which  is  not  reducible  farther  hy  x—b,  since  the  division 
does  not  terminate  without  a  remainder,  as  upon  trial  will  be 
found. 

Ex.6.  Reduce -TT — ^  V,^ r-r- —-to  its  lowest  terms. 

a^b  +  20*^/2  ^  2a?¥  -\-  a^b^ 

Here,  the  greatest  simple  divisor  of  the  numerator  and  de- 
nominator  is  evidently,  a^b  ;  Now, ^ =zdc^ 

cro 


ALGEBRAIC   FRACTIONS.  81 

ab^-\-b^.     Hence  the  result  is    »  .  »  ^7  ,  »  19  .  79  ;  and  the 

a^+2a^b-\-2ab^-\-b^ 

greatest  common  measure  of  this  result  is  a-{-bj  which  is  found 

thus; 

a^-\-2a^-\-2ab^'\-b^)  da^+lOd^b-^dab^S 

da^+lOa^+lOab^+ob^ 


remainder    ....    —5ab^  —  bb^ 


5aJ2 5^3 

And — =za-\-b,  which   by  another  operation  is 

—  50^ 

found  to  divide  the  numerator  without  a  remainder ;  and  con- 
sequently dividing  both  the  numerator  and  denominator  of  the 

fraction   — -, ,.,  ,  ,„    by  a-\-b,  we  have  the  fraction 

a^-{-2a^b-{-2ab'^-{-o'^ 

m  Its  lowest  terms;    that  is,  —7 =5a^+5a6; 

a^+2^b+2ab''+b^       .  ,     7  ,  ,2. 

and — =a^-\-ab  +  b^: 

a-\-b 

Hence  -^^^ — r-T-r7  is  the  fraction  in  its  lowest  terms. 
a^-}-ab-^b^ 

^      „    ^    ,         14a:V— 21iKV  .    •.    i 

Ex.  7.  Reduce <^ — ~-  to  its  lowest  terms. 

X 

Ex.  8.  Reduce to  its  lowest  terms. 

17a? 

3a;2-a?+2 
Ans. . 

a* 

Ex.  9.  Reduce    ,  .  ,,  to  its  lowest  terms. 

Ans 


a^^ab+b^ 


<v>4   I   fflcjip>  I    Q^ 

Ex.  10.  Reduce  —- = to  its  lowest  terms. 

a;* + ax^ — o?x — a* 

x^—ax-Va^ 
Ans. 


x'-—a'- 


Ex.  11.  Reduce  -^.^lr?M+|.^  to  its  lowest  terms. 

la— 2b 


ALGEBRAIC  FRACTIONS. 


a* J4 

Ex.  12.  Reduce  — — —  to  its  lowest  terms. 


Ans.       «'+*= 


y4 ^4 

Ex.  13.  Reduce  — ^ to  its  lowest  terms. 

Ans.  ^— — . 

Ex.  14.  Reduce — — rr^rr^^j to  its  lowest  terras. 

a*4-a2^2_|_^4 

«  —  b 
Ans 


Ex.  15.  Reduce to  its  lowest  terms. 


Ans. 


a^+ab-i-b^' 
!st  terms. 


Ex.  16.  Reduce -— — -7-- — -rr-r  to  its  lowest  terms. 
2a'^—3ba^—5b^a^ 

Ans. 


a-\-x 
jrms. 
a+2&+362 


2a2~3^>a— 562* 
a  J — a^x — ax^ + ic^ 


Ex.  17.  Reduce  — ;,-; — -  to  its  lowest  terms. 


Ans. 


a-\-x 

Ex.  18.  Reduce    _  .  ^  ,  ,  ,^  to  its  lowest  terms. 
a^'^2ab-\-b^ 

.        a^—ab 
Ans.  — r-v- 
a-\-b 

CASE  IV. 

TV?  reduee  fractions  to  other  equivalent  ones^  that  shall 
have  a  common  denominator. 

RULE  I. 

151.  Multiply  each  of  the  numerators  separately,  into  all 
the  denominators,  except  its  own,  for  the  new  numerators,  and 
all  the  denominators  together  for  the  common  denominator. 

It  is  necessary  to  remark,  that,  if  there  are  whole  or  mixed 
quantities,  they  must  be  reduced  to  improper  fractions,  and  then 
proceed  according  to  the  rule. 


ALGEBRAIC  FRACTIONS.  83 

Ex.  1.  Reduce  -— ,  —  and  -  to  a  common  denominator. 
4      c  a 


3aXcXa  =  3a^c 

5bX4x  a=z20ab  }  new  numerators  ; 
xxcx 


Xa  =  3a^c  J 
Xa=20ab  V 
X4:  =  4:cx    y 


4  X  c  X  a=4ac  common  denominator ; 

TT  .!_     r      .•  J         Sa^c    20ab        .  4cx 

Hence  the  fractions  required  are  - — ,  — — ,  and  . 

4ac      4ac  4ac 

Ex.  2.  Reduce  — — j — ,  and  to  a  common  denominator. 

3b  X 

{2x-\-\)Xx  =z2a:24.a:> 

^       2Jx3b  =  e>a^b       J  "ew  numerators  ; 

3b  X   x = 35a;  common  denominator  ; 

2a;^4-a?  Qa^b 

Hence  the  fractions  required  are  — — ; ,  and    -, — . 

36a?  3bx 

Ex.  3.  geduce  -,  — -,  and  a-\ — —  to  a  common  denomina- 

tor. 

.  3a:2      5a4-3a:2 
Here  a-\- 


5  5 

3x3x5z=:45 
5a:  X  4  X  5  r=  1 00a?  J-  new  numerator ; 

(5<z+3a;2)x4x3=60a+36a:2 


4  X  3  X  5  =  60  common  denominator  ; 

„  .u    f      .•  •    J         45    100a?        -  60a+36a;2 

Hence  the  tractions  required  are  ^-r,  -ttt-,  and ~- . 

oO      oO  60 


RULE  II. 

152.  Find  the  least  common  multiple  of  all  the  denomina- 
tors of  the  given  fractions,  (Art.  147),  and  it  will  be  the  com- 
mon denominator  required. 

Divide  the  common  denominator  by  the  denominator  of 
each  fraction,  separately,  and  multiply  the  quotient  by  the  re- 
spective numerators,  and  the  products  will  be  the  numerators 
of  the  fractions  required. 

Ex.  4.  Reduce and  — ^  to  the  least  common  denomi- 

aj2  4aa;2 

nator. 


84  ALGEBRAIC  FRACTIONS. 

Here   Aax^  is  the  least  common  multiple  of  x^  and  AaaP^ ; 
then  —^X^a:^b=A:aX^a'^h=zl2a?h 

4^^2  I  new  numerators. 

ajad  - — -x^ab=L5ah 

4ax^ 

iience  — — —  and  - — -  are  the  fractions  required. 
4ax'^  Aax^ 

Or,  as  4aa;2  (the  least  common  multiple)  is  the   denomina- 
tor of  one  of  the  fractions,  it  is  only  necessary  to  reduce  the 

fraction  — ~  ^o  an  equivalent  one,  whose  denominator  shall 

.       .     o      1  4r/.r2  Za^h       4a     3a^bx4a 

be  4ax'^  ;    hence,  — -—  =r  4a,  and 


a;'*  x^         4a       x^x4a 

— — 5  is  the  fraction  required. 

These  rules  appear  evident  from  (Art.  118).  For,  let 
T-,  ^,  7-  be  the  proposed  fractions  ;  then  7—,  7—,  ^-r-.,  are  frac- 
tions of  the  same  value  with  the  former,  having  the  common 
denominator  idf.     Since  ^=| ;  ^^=^  ;  and  f|.==l. 

3a^b      V  5x'^ 

Ex.  5.  Reduce r-,  ^,  and  r — x  to  the  least  common  de- 

4cx^    2x  8ac^ 

nominator. 

Here,  the  least   common   multiple  of  4cx'^,  2x,  and  8ac', 

(Art.  147),  is  Sac^x^  ;  then, 

-^^  X  3a2i = 2ac  X  3a^b  —  Qa^c 

8ac^x'^ 


Xi/=4ac^xXy=:4ac^xy 


2x 

Sac^x"^ 

— -—- X  5a:2  -  a;2  X  5^2  =  5a;* 
8flc2 


i-new  numerators ; 


Hence  - — r-^,  - — ~,  and  - — ^r-^  are  the  fractions  required. 

Ex.  6.  Reduce  — ; — ,  -— — >  and  :r-  to  a  common  denomi- 
a+a?       3    '  2x 


nator. 


30a;2        2a'^x—2x^  3a-f3ar 

6aa:-f6^'  6ax  + 6x2  '  ?"^  6aa:+6«2- 


ALGEBRAIC  FRACTIONS.  85 

T7    ^    n   J        2a?+3        ,  5a;4-2  , 

JiiX.  7.  Keduce ,  and     ^  ,     to  a  common  denomi- 

X  Sab 


nator. 


.         6abx-\-9ah         .  5a;2+2a; 

Ans.  — -r ,  and  —7—7 —  . 

3aox  3aox 

Ex.  8.  Reduce  -,  — - — ,  and  -— ; —  to  a  common  denomi- 
3      4  1-f-a? 


nator. 


4a?2+4a;    3x^+6a;+3  12at— 12 

^''^-  12a;+12'      m-hl2    '  *'''*  l2^+12' 

g  20^^  3a a?2 

Ex.  9.  Reduce  t,  -1-,  and  x-\ to  a  common  deno- 

0    a  X 

minator. 

adx    2hc'^x        ,  Zahd 
bdx  '    bdx  '  .         bdx 

17       in    r)   J         a;2  a;2_5  4flf~15 

liiX.  10.  Keduce  -—-,  a — ,  and  74 ;; —  to  a  com- 

5y  3a?  2 

mon  denominator. 

6x^    30axi/  —  l0x^i/-\-50j/  60axy — 15a;y 

■  SOxy"  30xy  '  ^"  30^y 

Ex.  1 1 .  Reduce  — — -, -— ,  and  — - —  to  other  equiva- 

a^—x^   4a— 4x  a-\-x 

lent  fractions  having  the  least  common  denominator. 

4a         3ab-\-3bx         ,  20ax--20x^ 
Ans.    .'     ■    .  „,  — -7; — — r,  and 


4a2_4a,2'    4o2_4a;2'    .  4a2_4a;2    * 

Ex.  12.  Reduce  -— — — ^,  — r,  and  -r^-r-    to   the 

a24-2aa;-|-a;2   a^—x^  a*— a;* 

least  common  denominator. 

a3_^aa;2— aV— a:^     a^-\-ax'^-^a^x-{-ci^ 

and    ,    ^"3^+^^^^     ,. 
o^— aiK*+a*a;— a* 


§  V.    ADDITION  AND    SUBTRACTION  OF  ALGEBRAIC  FRACTIONS. 

To  add  fractional  quantities  together. 

RULE. 

153.  Reduce  the  fractions,  if  necessary,  to  a  common  deno- 
minator, by  the  rules  in  the  last  case,  then  add  all  the  nume- 

9 


86  ALGEBRAIC  FRACTIONS. 

rators  together,  and  under  their  sum  put  the  common  denomi- 
nator ;  bring  the  resulting  fraction  to  its  lowest  terms,  and  it 
will  be  the  sum  required. 

Ex.  1 .  Add  — ,  — ,  and  -  together. 

2a;X  7X9  =  126a; 
5a;X3x9  =  135a? 


a;X7x8=:  21a: 
3X7X9=189 


126a;-i-135x-l-21a?     282a? 


189  189 

93a; 
+  -—  is  the  sum  required. 


Ex.  2.  Add  -,  -y,  and  — -  together. 
0    So  4a 

l2a^b  +  8a^+15P 


12a62 

aXBhx4a=l2a^^20aH+l5b^       ,^.  .^.       ^     ,, 
2aXbx4a=  Sa'b  \ r^-Ti =  (dividing  by  b) 

5bx3bxb=15b^     I  ^^^3 

—  I ; IS  the  sum  required. 

bx3bx4a=:l2ab^}        1^«^ 

Or,  the  least  common  multiple  of  the  denominators  may  be 
found,  and  then  proceed,  as  in  (Art.  152). 

It  is  generally  understood  that  mixed  quantities  are  reduced 
to  improper  fractions,  before  we  perform  any  of  the  operations 
of  Addition  and  Subtraction.  But  it  is  best  to  bring  the  frac- 
tional parts  only  to  a  common  denominator,  and  to  affix  their 
sum  or  difference  to  the  sum  or  difference  of  the  integral  parts, 

interposing  the  proper  sign. 

33.2 
Ex.  3.  It  is  required  to  find  the  sum  of  a ^-,   and   b-\- 

2ax 
c 

3j;2      ab—Boc^        ,  ,  .  2aa;      bc+2ax 

Here,  a j— = 1 — >  ^^^  o-\ = . 

00  c  c 

Then,  (ab-3x^)Xc=abc-3cx'^  Enumerators 
(bc+2ax)  X  b  =  b^c+2abx  ]  «"n^«^^^o^«- 

bXc=bcz=  denominator. 

'abc—3cx'^-\-b'^c-\-2abx_aI}C-^b^c 
be  be 

2abx  —  3cx^        ,  ,  ,  2abx—3cx^ 

— fc — =«+*+— j^ — 

is  the  sum  required. 


ALGEBRAIC  FRACTIONS.  87 

Or,  bringing  the  fractional  parts  only  to  a  common  deno- 
minator, 

Thus,  3x'^xc=3cx^  ) 

And  5  Xcr=5c  common  denominator. 

„,,  3c^2  ^5jc  2a5a?— 3ca:2  , 

Whence  a ; \-b'\ — - — =a-\-b-{ j the  sum. 

Oc  be  be 


Ex.  4.  It  is  required  to  find  tho  sura  of  5pc-{ — -—  and  4« 
2a:-3 


5a3 

Here,  (  a;— 2)  x  5a; =5*2— 10a;  )  „„^^^,^,, 
/o  o\vxo  c  n  >  numerators, 
(2ir--3)x3  =6a;  —9      >  ' 


And  3  x5a;  =  15a;  common  denominator. 

Whence  5a;i — h4a; — =9a;-| ^ 

15a;  15a;     .  15a; 

9--6a;     ^     .  5a;2— 16a;4-9    ,  .  ' 

— — — =9a;H — the  sum  required. 

i  037  loa; 

g-ji 9  9 6-j. 

Here, -— —  is  evidently  =  — — —  (Art.  128) .;  but  we 

loa;  loa; 

might  change  the  fractions  into  other  equivalent  forms  before 

we  begin  to  add  or  subtract ;  thus,  the  fractional  part  of  the 

2a; 3 

proposed  quantity  4a; maybe  transformed  by  chang- 

tax 

ing  the  signs  of  the  numerator,  (Art.  128),  and  the  quantity 

3 2x 

itself  can  be  written  thus,  4a; H —  ;  It  is  well  to  keep  this* 

oa; 

transformation  in  mind,  as  it  is  often  necessary  to  make  use  of 
it  in  performing  several  algebraical  operations. 

T7      .:     A  J  ,  3«2   2«       ,  6 

Ex.  5    Add  — ,  —  and  -  together. 

,         105a2  4-28a5+10J2 

^""  70^ 

Ex.  6.  Add and  — — -  together.  Ans.  -r — ;r. 

a;— 3  a;+3  a;^— 9 

Ex.  7.  Add  ^ — ~  and  ^^^  together. 
a~b  a+b      ^ 

2a^+2b^ 
Ans.  . 

a^-^b^ 


88  ALGEBRAIC  FRACTIONS. 

Ex.  8.  Add and  —  '^—^  together. 

a — X  a-^-x     ° 


*  4cfa? 

Ans. 


Ex.  9.   Add  2a;+^  and  3a;^?^  together. 


a^—x^ 


Ans.  5a: + 


12 


Ex.  10.  Add  4a;,  -—  and  2+^  together. 


Ans.  4a;+24-^^. 
45 

Ex.  11.  Add  5ap —  and  — 4a?  together 

Ans.  0.+— . 

Ex.  12.  It  is  required    to  find  the  sum  of  2a,  — ^,  and 

a — X 


Ans.  2a+24— r — 


To  subtract  one  fractional  quantky  from  another. 

RULE. 

154.  Reduce  the  fractions  to  a  common  denominator,  if  ne- 
cessary, and  then  subtract  the  numerators  from  each  other, 
and  under  the  difference  write  the  common  denominator,  and 
it  will  give  the  difference  of  the  fractions  required. 

Or,  enclose  the  fractional  quantity  to  be  subtracted  in  a 
parentheses  ;  then,  prefixing  the  negative  sign,  and  perform- 
ing the  operation,  observing  the  same  remarks  and  rules  as  in 
addition,  the  result  will  be  the  difference  required. 

The  reason  of  this  is  evident ;  because,  adding  a  negative 

quantity  is  equivalent  to  subtracting  a  positive  one  (Art.  63) ; 

thus,  prefixing  the  negative  sign  to  the   fractional  quantity 

a  —  h     .     .  /a  —  b\  a  —  h        h — a  . 

,  It    becomes  — -  i )  = =  ;    to   the 

c  \    c    J  c  c 

Qc    I  fit                                  y      X  .li ,  ftK 
fractional   quantity    —    ,  it  becomes    —  f ^^j  = 

,  x^-\-a  ,.        ,^„.            1     /.       .       1           .           ox — h    . 
•T (Art.   128)^  to  the  fractional  quantity — ,  it 


ALGEBRAIC  FRACTIONS.  89 

becomes  —  ( J=  — - — ;    to   the    mixed    quantity 

y  \  y    f  y 

2 X 

and   to  the   mijjed  quantity  —  3a+  ,   it  becomes   — 

Ex.  1.  Subtract —,  from -^. 
Here  3a;x7=21a:>    ,  ^^^,,^,„ 


5  X  7r=35  com.  denom.* 

Ex.  2.  Subtract  — from  -t-t-- 

5c  3d 

Here  (2a-4.)  X  3J=6a6-12Ja. )  „„„,^^„. 

(x—y)X5c=       5cx—  5ci/ S 


25a;— 21a;      4x.    . 

.'. =  -^-is  the 

35  35 

difference  required. 


5cx35  =  15&c  common  denominator. 

,  5ox—5cy     Gab  —  125a;     Scar— 5cy     125a; — 6ab 

^^^"''^'  "Tsi^  15b^~  "^      I5bc     "^       156^ 

5ca;— 5cv4-126a;— 6n5  .     ,      ,._.  .     , 

-^—i IS  the  difference  required. 

lo6c 

^     .           n  •       -,              '         '             1               .      2a — 4x 
Or,  by  prefixing  the  negative  sign  to  the  quantity  — , 

DC 

.  ,  2a— 4a;       4x — 2a      ,        .        ,  •      .       ij 

it  becomes = ;  then  it  only  remains  to  add 

5c  5c  ^ 

— and      -T  together,  as  in  addition,  .and  the  result  will 

5c  So 

be  the  same  as  above. 

Ex.  3.  From  2a5H ; — subtract  2a5 


ayf-a;  a-{-x 

Here  prefixing   the  tiegative    sign  to  the  quantity  2a5— 

— ; — ,  we  have  —  (2a5 )  =  — 2a5H ;  hence  the 

a-{-x  \  a-\-x/  a-\-x 

difference  of  the  proposed  fractions  is  equivalent  to  the  sum 

of  2a6-f  ^^^,  and  —2ab-{-  ^— —  ;  but  the  sum  of  the  frac- 
a-\-x  a—x 

9* 


90  ALGEBRAIC  FRACTIONS. 

-         ^    a—x       .  a-\-x   .    2a24-2a;2 

tional  parts  — ; —  and ,  is  — —  :  Therefore  the  diffe- 

a-{'X         a—x         a^—x^ 

.     :,  •    ^  X     o  1  .    2a2+2a;^     2a2+2a;2 
rence  required  is  2ab—2ab-\ ^ T  —  — 2 2~* 

Ex.  4.  From subtract  — - — . 

15  7 

Here  (I0a;-9)X    7=70a:-63  >    „^^    .^^^      . 

(3a:-5)xl5  =  45x-75r""'''''''''- 


15x7=105  common  denominator. 

^,       ^        70a;— 63       45a;-75 
Therefore, 


105  105 

70a;— 63-45a;+75     25a;+  12 


105  105 


is  the  fraction  required. 


_       ^    _,        a-\-h      .  a—h  .  Aah 

Ex.  5.  From subtract     .     ■.  Ans. 


a—b  a-\-b  a^ — b"^ 

Ex.  6.  From subtract  — ; —  Ans. 


;2— .i?2 


a — X  fl+a?  (v" — X' 

_      ^    _        4a;+2      .         ^  2a;— 3  .        4a:2+3 

Ex.  7.  From  — - —  subtract  — - — .  Ans. 


3a;  3a; 


Ex.  8.  From  3a;  +  y  subtract  x— 


b c  ' 

cx-\-bx — ab 


Ans.  2a;+ 


bc 


^      ^    o  ,  2a;+7  ^        3x'^+a^ 

Ex.  9.  Subtract — ?-- from        ^ 


8  3b 

24a;2+8a2— 6ia;— 2U 


Ans. 


24^> 


T,      ,«    «  1           .        2a;— 3  -        ^     ,   a;— 2 
Ex.  10.  Subtract  4a; —  from  5a; -i — - 


Ans.  x-i r-z — . 

15 


Ex.  11.  Subtract -T rfroma4- 


a{a—x)  a^a-^-x)' 

4x 
Ans.  a— 


Ex.  12.  Required  he  difference  of  3a;  and 


a^—x'^' 

3g+12a; 

5 

3a;— 3a 
Ans.  — r 


ALGEBRAIC  FRACTIONS.  91 


Ex.  13.  From2a;+-^- — subtract  3a; ^J-. 


.        16ir+23 
Ans.-^^ 


^  VI.     MULTIPLICATION  AND  DIVISION  OF  ALGEBRAIC  FRAC- 
TIONS. 

To  multiply  fractional  quantities  together. 

RULE. 

155.  Multiply  their  numerators  together  for  a  new  nume- 
rator, and  their  denominators  together  for  a  new  denominator  ; 
reduce  the  resulting  fraction  to  its  lowest  terms,  and  it  will  be 
the  product  of  the  fractions  required. 

It  has  been  already  observed,  (Art.  119),  that  when  a  frac- 
tion is  to  be  multiplied  by  a  whole  quantity,  the  numerator 
is  multiplied  by  that  quantity,  and  the  denominator  is  retain- 
ed : 

rrn         f"'  ac       ,  2a?     ^      10a?  i  •  t    .      i 

Ihus,  yXc=-j-,  and-r-x5=— 7—  ;  or,  which  is  the  same, 
bob  b 

making  an  improper  fraction  of  the  integral  quantity,   and 
then  proceeding  according  to  the  rule,  we  have  tXt-=-t-, 
,  2a;     5     10a; 

Hence,  if  a  fraction  be  multiplied  by  its  denominator,  the 

product  is  the  numerator ;    thus,  ^xh  =i-r  =zh.      In   like 

b  b 

manner,  the  result  being  the  same,  whether  the  numerator 
be  multiplied  by  a  whole  quantity,  or  the  denominator  divid- 
ed by  it,  the  latter  method  is  to  be  preferred,  when  the  de- 
nominator is  some   multiple  of  the    multiplier  :     Thus,  let 

— -  be  the  fraction,  and  c  the  multiplier  ;  then^— Xcr=:-T— = 
he  ,  ^  be  be 

ad  ad  ad       ad         ,    ^ 

-rr  ;  and  ^-  X  c= =-i-,  as  before. 

b  be  bc-^c      b 

Also,  when  the  numerator  of  one  of  the  fractions  to  be  mul- 
tiplied, and  the  denominator  of  the  other,  can  be  divided  by 
some  quantity  -which  is  common  to  each  of  them,  the  quo- 
tients may  be  used  instead  of  the  fractions  themselves  ;  thus 


92  ALGEBRAIC  FRACTIONS. 

a-{-b         X  X 

rX — — ,  = J  ;  cancelling  x-\-h  in  the  numerator  of  the 

a—b     a-\-b     a—b  ^ 

one*  and  denominator  of  the  other. 
Ex.  1.  Muhiply^by  ^. 

3aX4a=12a2—  numerator,  )  *i,     r     *•  •    j  • 

5  X  7=35=  denominator  ,'  \         '''  '^^  ^^'^^^^^  ^^^^^^^  ^^ 

12a2 

35  • 
Ex.2.  Multiply?^  by  y. 

Here,  (3x4-2)  X  8a;=i24a;2-j-  16a;=:  numerator, 
and  4x7=28=  denominator  ; 
24a;2-{-  16a: 
Therefore,  -— —  =  (dividing  the  numerator  and  de- 

28 

nominator  by  4) — ,  the  product  required. 

^2 ^2  '7/p2 

Ex.  3.  Multiply  — - —  by 


3a       -^  a—x 

Here,  (a^— x^)  X  lx'^=(a-\-x)  x  (a—x)x  '7x'^=  numerator 

(Art.  106),  and  3ax(a—x)=  denominator  ;  see  Ex.  15,  (Art. 

79.) 

,        .     (a-\-x)  X  {a—x)  X  7a;2  . 
Hence,  the  product  is  ^ '- — ^ =  (dividing 

7a;2  /q  _|_  ^\ 

the   numerator   and  denominator   by   a—x,) = 

7ax^+7x3 
3a       ' 

Ex.  4.  Multiply  a+^  by  a—- 

X     5a+x       -        X     3a — x 
Here,a+-=-— ,  and  a^-=-^  : 

Then,  (5a4-a;)x(3a— x)r=15a2— 2ax— x2=  new  numerator, 

rm        r         ISa^ — 2ax— x^ 
and  5x3  =  15=  denominator:  Therefore,  - 


15 


2  ftdc  "T"  jc 

is  the  product  required. 


15 

156.  But,  when  mixed  quantities  are  to  be  multiplied  to- 
gether, it  is  sometimes  more  convenient  to  proceed,  as  in  the 
multiplication  of  integral  quantities,  without  reducing  them  to 
improper  fractions. 


ALGEBRAIC  FRACTIONS.  93 

Ex.  5-  Multiply  a;2- ia;+|  by  Ja:+2. 

J^H-2 

t 


3-^ 


■i^+i 


Ex.  6.  Multiply  — — by 


Ans. 


14  -^  3a;3-3a; 

3ax — 5a 

t:^      .r    Tv/r  n-  i         3a;2  15a?-30  ^        9a: 

Ex.  7.  Multiply  3—^  by  -^^.  Ans.  _. 

■I?      o    TIT  u-  1     2a--2a?  ,         3ax  ^        2x 

Ex.  8.  Muluply  -^^  by  3^^-.  Ans.  ^. 

Ex.  9.  It  is  required  to   find   the    continual   product   of 
3a   2x^        ,  a~^b                                                 ^         2ax+2bx 
-5-'  -3-'  ^""^  ~^'  ^"^-  5 ' 

Ex.  10.  It  is  required  to  find  the   continued   product  of 
a*— a?*     a-\-y         .  a—y 

Ex.  11.  It  is  required  to  find   the  continued  product  of 
a2_a.?  ^2_^2  ^  a2_3^, 

— pv-,  — :-- ,  and -.  Ans.  . 

a+o      a-\-x  ax—x^  x 

Ex.  12.  Multiply  a?2—|a?+l  by  a;2— ^a;. 

Ans.  a:^— 5a;34-ya:2— Jar 


To  divide  one  fractional  quantity  by  another. 

RULE. 

157.  Multiply  the  dividend  by  the  reciprocal  of  the  divisor, 
or  which  is  the  same,  invert  the  divisor,  and  proceed,  in  every 
respect,  as. in  multiplication  of  algebraic  fractions;  and  the 
product  thus  found  will  be  the  quotient  required. 

When  a  fraction  is  to  be  divided  by  an  integral  quantity ; 
the  process  is  the  reverse  of  that  in  multiplication  ;  or,  which 
is  the  same,  multiply  the  denominator  by  the  integral,  (Art. 
120),  or  divide  the  numerator  by  it.  The  latter  mode  is  to  be 
preferred,  when  the  numerator  is  a  multiple  of  the  divisor. 


94  ALGEBRAIC  FRACTIONS. 

Ex.  1.  Divide — by-. 
a     ''  c 

u  c  ^t*     c       5coc 

The  divisor  -  inverted,  becomes  y.  hence  — Xt  =  —t   is 
c  b  a      0       ab 

the  fraction  required. 

T-.      ^    T^-  -1     3a  — 3a:  ,      5a— 5a? 

Ex.  2.  Divide ;—  by  — -r— . 

a-\-b       ^     a-\-b 

The  divisor  ( — —: — )  inverted,  becomes 
\  a-\-b    I 


5a — 5a; 


,  3a— 3a?       a-i-b        3a  —  3a;     3(a — x)     3    .       , 

hence tt-Xt j-=^ T"— ^7 \='^  ^^   t^®   ^^^' 

a-^-b       5a  — 5x      5a— 5x     5[a—x)     5 

tient  required. 

Ex.  3.  Divide  ^  "~     by  a+b. 

X 

I  (j^ ^2  \ 

The  reciprocal  of  the  divisor  is  — — r  ;  hence ^X  — — r 

^  a-\-b  X         a-\-b 

(a4-^)(«  — ^)     a—b  .     .  .     ^         .     , 

z=:- p — M       ^®  *"®  quotient  required. 

Or, ~z=a—b  ;  hence  is  the  fraction  required. 

a-f-6  X 

Ex.4.  Divide  — ; — by  oH . 

a+c  .       « 

.  a:2-.a2     a2_^^2_«2     a;2.    ,        .,/..•      a;^— a2 

Here,  a-\ = — =— ;  then,  the  fraction — ; — 

a   .  a  a  a-tc 

<ip2  <yi2 fji  n  '    dX^  ~—  a 

divided  by  —  becomes  — ; — X--= — r=  the  quotient 

•'     a  .     a-\-c       x^     ax^-\-cx^ 

required. 

158.  But  it  is,  however,  frequently  more   simple  in  prac- 
tice to  divide   mixed  quantities  by  one  another,  without  re- 
ducing them  to  improper  fractions,  as  in   division  of  integral 
quantities,  especially  when  the  division  would  terminate. 
Ex.  5.  Divide  x'^—\x^-\-^-^x'^—\x  by  x'^—\x. 
a;2— ix)a:*-{a;3+  ^■^x'^—^x{x'^-\x-\- 1 
a;*— ia;3 


-fc 

34. 

ya:2- 

-¥ 

^4- 

V 

a;2- 

-¥ 

x^- 

-¥ 

SIMPLE  EQUATIONS. 

Ex    6.  Divide  —  bv  ^. 
3     -     5 

Ex.  7.  Divide  — ~  Ijy  — -! — . 
5        "^      5a? 

•  Ex.  8.  Divide    ^~  — by~. 
5  o 

Ex.9.Dmde— 2^-^by^j-. 

2a;2  J, 

Ex.  10.  Divide -T-j — r-by — ; — .  Ans. 


a^-}-x^       x-{-a'  '  x^—ax-\-a^ 


Ex.  11.  Divide — ■ by 


a-\-x      "^  a'^-\-2aX'{-x^ 

Ans.  a^-\'2a^x+2ax^+x^» 

Ex.  12.  Divide  a;*—  ~sc3^x^+-x—2  by  ^a:-2. 

Ans.-a^— .-of^-J-l 
4         * 


CHAPTER  III. 

ON 

SIMPLE  EQUATIONS, 

INVOLVING  ONLY  ONE  UNKNOWN  aUANTITY. 

159.  In  addition  to  what  has  been  already  said,  (Art.  34), 
it  may  be  here  observed,  that  the  expression,  in  algebraic 
symbols,  of  two  equivalent  phrases  contained  in  the  enuncia- 
tion of  a  question,  is  called  an  equation,  which,  as  has  been 
remarked  by  Garnier,  differs  from  an  equality,  in  this,  that 
the  first  comprehends  an  unkn6\vn  quantity  combined  with 
certain  known  quantities  ;  whereas  the  second  takes  place 
but  between  quantities  that  are  known.     Thus,  the  expression 

s     d   . 
«=2^"2'  ^^^^*  ^^^^'  ^^^<^^^i^g  ^o  t^®  above  remark,  is  called 

an  equality ;  because  the  quantities  a,  ^,  and  d,  are  supposed 
to  be  known.  And  the  expression  x^x—d=s,  (Art.  103),  is 
called  an  equation,  because  the  unknown  quantity  x,  is  com- 
bined with  the  given  quantities  d  and  s.     Also,  a— a=0  is  an 


96  SIMPLE  EQUATIONS. 

equation  which  asserts  that  x--  a  is  equal  to  nothing,  and  there- 
fore, that  the  positive  part  of  the  expression  is  equal  to  the  ne- 
gative part. 

160.  A  simple  equation  is  thatw^ich  contains  only  the  first 
power  of  the  unknown  quantity,  or  the  unknown  quantity 
merely  in  its  simplest  form,  after  the  terms  of  the  equation  ha^e 
been  properly  arranged : 

X       X 

Thus,  x-\^a=zb  ;  ax-\-hx=c ;  or,  --{--=zdt  &c.  where  a?  de- 

4     o 

notes  the  unknown  quantity,  and  the  other  letters,  or  numbers, 

the  known  quantities. 


§  I.    REDUCTION  OF  SIMPLE  EQUATIONS. 

161.  Any  quantity  may  be  transposed  from  one  side  of  an  equa- 
tion to  the  other,  by  changing  its  sign. 

Because,  in  this  transposition,  the  same  quantity  is  merely 
added  to  or  subtracted  from  each  side  of  the  equation;  and, 
(Art.  48,  49,)  if  equals  be  added  to  or  subtracted  from  equal 
quantities,  the  sums  or  remainders  will  be  equal'.     Thus,  if  x 
+  5  =  12  ;  by  subtracting  5  from  each  side,  we  shall  have    ^ 
a;+5-5=z:12-5; 
but5  — 5=0,  and  12  — 5=7;  hence  a; =7. 
Also,  if  x-{-a=b^2x ;  by  subtracting  a  from  each  side,  we 
shall  have 

x-{-a — a=b—2cc—ai 
and  by  adding  2x  to  each  side,  we  shall  have 
x-\ra — a+2a;=^ — 2x — a-{-2x  ; 
hut  a—a=0,  and  — 2a;-f  2a;=0:  therefore 
x-\-2x=b—a,  or  3x=b — a. 
Again,  if  ax — c=:d,  and  c  be  added  to  each  side,  ax — c-f-c 
=d-{'C,  or  ax=d-{-c. 

Also,  if  5aj— 7=2a;+ 12  ;  by  subtracting  2x  from  each  side, 
we  shall  have 

5oo—7—2x=2x-{-l2—2x,  or  3a:— 7=12  ; 
subtracting  — 7,  or,  which  is  the  same  thing,  adding +7  to  each 
side  of  this  last  equation,  and  we  shall  have 
3a;— 7+7  =  12  +  7; 
but  7-7=0,  .•.3a;  =  19. 
Finally,  if  x—a+b=c—2x+d,  then,  by  subtracting  h  from 
each  side,  we  shall  have 

X — a-\-b — b=c—2x-\-d — b  ; 
and  adding  a-\-2x  to  each  side,  it  becomes 


SIMPLE  EQUATIONS.  97 

x—a+b—b+a-{-2x=c—2x-\-d—b-{-a-\-2x ; 
but  a  —  a=0,  b  —  b—0,  and  — 2a!-f  2a;=0  ; 
therefore,  x-\-2x—c-\-a^b-i-d,  or  3x=c-{-a—b-\-d. 

Cor.  1 .  Hence,  if  the  mgns  of  the  terms  on  each  side  of 
an  equation  be  changed,  the  two  sides  still  remain  equal ;  be- 
cause in  this  change  every  term  is  transposed  :  Thus,  if  — x 
4-6 — c=a — 9-\-x  ;  then,  x—b-^c—9 — a—x  ;  or,  which  is  the 
same  thing,  by  transposing  the  right-hand  side  to  the  left,  and 
the  reverse,  we  shall  have  9  —  a — x=:x — b-{-c. 

Cor.  2.  Hence,  when  the  known  and  unknown  quantities 
are  connected  in  an  equation  by  the  signs  +  or  — ,  they  may 
be  separated  by  transposing  the  known  quantities  to  one  side, 
and  the  unknown  to  the  other. 

Thus,if3a;— 9— a=:12  +  6— 4a;2;then,4a;2-l-3a;=a-f6+21. 

Also,  if  3x^-^2-i-.x=b—4x^'-3x* ;  then,  3x^-\-4xH  3^2+ 
x=b-{-2. 

Hence  also,  if  any  quantity  be  found  on  both  sides  of  an 
equation,  it  may  be  taken  away  from  each  ;  thus,  if  x-^-a^za 
+  5,  then  a:=5  ;  if  x  —  b:=c-\-d—b,  then  x=^c-\-d  ;  because, 
by  adding  i  to  each  side,  we  shall  have  x — Z>-f-6=cH-c? — b-{-b  ; 
but  b — 6r=0,  .•.x=c-\-d. 

162.  If  every  term  on  each  side  of  an  equation  be  multiplied  by 
the  same  quantity,  the  results  will  be  equal :  because,  in  multi- 
plying every  term  on  each  side  by  any  quantity,  the  valiJfe  of 
the  whole  side  is  multiplied  by  that  quantity  ;  and,  (Art.  50), 
if  equals  be  multiplied  by  the  same  quantity,  the  products  will 
be  equal. 

Thus,  if  x=:^-\-a,  then  6a?=30-|-6a,  by  multiplying  every 

X 

term  by  6.     An3,if -=4,  then,  multiplying  each  side  by  2, 
2 

we  have  -X2=4x2,  or  a;=8,  because,  (Art.  155),-  X2=x. 

Also,  if-  — 3=a  — i,  then,  by  multiplying  every  term  by  4, 

we  shall  have  x — 12= 4a— 46. 

•.      .  3 

Again,  if  2x —  --\-\z=zx  ;   then,  4a;  — 3 4-2 =20; ;  and  Ax — 

2a;i=3— 2,  or  2x—\. 

Cor.  1.  Hence,  an  equation  of  which  any  part  is  fraction- 
al, may  be  reduced  to  an  equation  expressed  in  integers,  by 
multiplying  every  term  by  the  denominator  of  the  fraction  ; 
but  if  there  be  more  fractions  than  one  in  the  given  equation, 
it  may  be  so  reduced  by  multiplying  every  term  by  the  pro- 
duct of  the  denominators,  or  by  the  least  common  multiple  of 
10_ 


98  SIMPLE   EQUATIONS. 

them ;  and  it  will  be  of  more  advantage,  to  multiply  by  the 
least  common  multiple,  as  then  the  equation  will  be  in  its 
lowest  terras. 

CC  *T  C£  ^ 

Let  2  +  q  +  I"-'-^  '  ^^®"'  ^^  every  term  be  multiplied  by 
24,  which  is  the  product  of  all  the  denominators  ;  we  have 
^X24  +  ^X24-1-^X24  =  11X24;   and  12a;+8a;+6a?=264  ; 

"^  d  4 

or,  if  every  term  of  the  proposed  equation  be  multiplied  by 

12,  which  is  the  least  common  multiple  of  2,  3,  4,  (Art.  146) ; 

we  shall  have   6a:+4:c  + 30?=  132,  an  equation  in  its  lowest 

terms. 

Cor.  2.  Hence  also,  if  every  term  on  both  sides  have  a 

common  divisor,  that  common  divisor  may  be  taken  away ; 

,        ..3a;  ,  a4-6     2x-\-1    .  ,  .  ,   .  ^     , 

thus,  It  — -j — r— = — r — ,  then,  multiplymg  every  term  by  5, 
o         o  o 

we  shall  have  dx-\-a-\-Q—2x+l,  ox  x=zl—a. 

Also,  if 1 — = ,  then  multiplying  by  c,  we  shall 

have  ax — b-\-2=:7 — a?,  or  Ga?+iP=^+4. 

163.  If  every  term  on  each  side  of  an  equation  he  divided  hy 
the  same  quantity,  the  results  will  be  equal :  Because,  by  divid- 
ing every  term  on  each  side  by  any  quantity,  the  value  of  the 
whole  side  is  divided  by  that  quantity;  and,  (Art.  51),  if 
equals  be  divided  by  the  same  quantity,  the  products  will  be 
equal. 

Thus,  if  6a'^+3x=z9',  then,  dividing  by  3, 2o2-f  a?=3. 

Also,  if  ax^-\-bx=acx;  then,  dividing  every  term  by  the 

common  multiplier  x,  we  shall  have 4- — = — ,  or  ax-jro 

^  XXX 

=ac. 

Cor.  1.  Hence,  if  every  term  on  both  sides  have  a  common 
multiplier,  that  common  multiplier  inay  be  taken  away. 

Thus,  if  ax-{-ad=ab,  then,  dividing  every  term  by  the  com- 
mon multiplier  a,  we  shall  have  x-{-d  =  b. 

(IOC      (lb      4fl  3? 

Also,  if 1 — = ;  then  dividing  by  th€  common  multi- 

c       c         c 

plier  -,  or  (which  is  the  same  thing)  mwltiplying  by  -,  we  shall 

have  x-\-b=zAax. 

Cor.  2.  Also,  if  each  member  of  the  equation  have  a  com- 
mon divisor,  the  equation  may  be  reduced  by  dividing  both 
Bides,  by  that  common  divisor. 


SIMPLE  EQUATIONS.  99 

Thus,  Uax^-~a'^x=abx — a^b,  or  {ax~a'^)x={ax—a'^)b  ;  then 

it  is  evident  that  each  side  is  divisible  by  ax — a^,  whence  x=b. 

Again,  if  x"^  —  a'^=:x-\-a  ;  then,  because  x"^  —  a^=(x-\-a) 

.  {x—a),  it  is  evident  that  each  side  is  divisible  by  x-{-a  ;  and 

hence  we  have = — i — ,  or  x—a=l,  and  x=a-{-l. 

x-f-a      x-jra 

164.  The  unlinown  quantity  may  be  disengaged  from  a  divi- 
sor or  a  coejficient,  by  multiplying  or  dividing  all  the  terms  of 
the  equation  by  that  divisor  or  coejficient. 

Thus,  if  2a;+4=5,  then  x-\-2=     and  a;=-  —2. 

2  2t 

X 

Also,  let  -4-9=17  ;  then,  multiplying  by  2,  we  shall  have 

|x2+18  =  17X2, 
4> 

or  a;  +  18  =  34,  .-.  a;=34  — 18. 

Again,  let  ax-\-bx:=c—d,  or,  which  is  the  same,  let  {a-\-b)x 

=c—d  ;  then,  dividing  both  sides  by  a+^,  the  coefficient  of 

X,  and  we  shall  have 

c — d 

X=z -. 

a+b 

Finally,  let Y=c-\-d  ;  then,  the  equation  may  be  put 

under  this  form, 


\a       bJ' 


■c+d; 


and  dividing  each  side  by j-,  we  shall  have  x={c-{-d)-r' 

I t)  ;  which  may  be  still  farther  reduced,  because j- 

= — 7—  ;  therefore 
ao 

or  x=(c+d)x  7 , 

0  —  a 

abc-\-abd 

.•.X=z — . 

0  —  a 

165.  Any  proportion  may  be  converted  into  an  equation  ;  for 
the  product  of  the  extremes  is  equal  to  the  product  of  the  means. 


100  SIMPLE  EQUATIONS 

Because,  i(  a  :  b  : :  x  :  d ;  then  t=  jj  (Art.  24),  and  .*. 

(Art.  162),  ad=zbx,  by  clearing  effractions. 

Let  3a;  :  5a:  :  :  2a;  :  7  ;  then  7x3x=2xX5x, 

or  21a:=10a;2  :  and  .•.21=:10a;. 

Again,  let  5a;+20  :  4x+4:  :  :  5  :  a;+]  ;  then, 

(5X+20)  x{x-hl)  =  5x  (4a:+4)  ; 

or,  5a;2+25a;+20=20a;  +  20  ; 

and  (Art.  161),  5a;24-25a;=20a; ; 

.-.(Art.  163),  5a:+25=20r 

166.  When  an  unknown  quantity  enters  into,  or  forms  a 
part  of  an  equation  ;  and  if  the  equation  can  be  so  ordered, 
that  the  unknown  quantity  may  stand  by  itself  on.  one  side, 
with  its  simple  or  first  power,  and  only  known  quantities  on 
the  other,  the  quantity  that  was  before  unknown,  will  then 
become  known. 

Thus,  suppose   3a;-|-18=:5a; — 2;  then,  by  transposing  3a; 

and  — 2,  we  shall  have 

184-2=5a;— 3a;,  or  20=:2a; ; 

20      ,^ 
thereiore,  a;=r— =10. 

Here,  in  the  above  equation,  the  value  of  the  unknown 
quantity  a;,  becomes  known,  and  10  is  the  value  of  x  that  ful- 
fils the  condition  required,  which  we  can  readily  see  verified, 
by  substituting  this  value  of  x  in  the  givjen  equation  ;  thus, 

3a;=3  X  10  =  30,  and  5a;  =  5  X  10=50  ; 
hence,  3a;+ 18  =  30+18  =  48,  and  5a;— 2  =  50  —  2  =  48  ; 
therefore  10  is  the  true  value  of  x,  which  answers  the  condi- 
tion required,  and  this  value  of  x  is  called  the  root  of  the  equa- 
tion. 

167.  Hence  the  root  of  an  equation  is  such  a  number  or 
quantity,  as,  being  substituted  for  the  unknown  quantity,  will 
make  both  sides  of  the  equation  vanish  or  equal  to  each  other : 
Thus,  in  the  simple  equation 

3a;— 9  +  6=0; 
the  value  of  x  must  be  such,  that  if  substituted  for  it,  both 
sides  must  vanish,  because  the  right-hand  side  is  0  ;  but  this 
value  is  found  to  be  1,  for  by  transposition 

3a;  =  9  — 6=3, 

and  dividing  by  3,  we  shall  have 

3a;     3 
-=-,ora.=  l; 

therefore  1  is  the  root  of  the   given  equation,  which  can  be 
easily  verified  by  substituting  it  for  x  ;  thus, 


SIMPLE  EQUATIONS.  101 

3a;— 9  +  6  =  3x1—9  +  6  =  3— 94-6=9— 9=0. 
Hence,  the  value  of  the  unknown  quantity  being  substitu- 
ted in  the  equation,  will  always  reduce  it  to  0=0. 

^  II.     RESOLUTION    OF    SIMPLE    EQUATIONS, 

Involving  only  one  unknown  Quantity . 

168.  The  resolution  of  simple  equations  is  the  disengaging 
of  the  unknown  quantity,  in  all  such  expressions,  from  the 
other  quantities  with  which  it  is  connected ;  and  making  it 
stand  alone,  on  one  side  of  the  equation,  so  as  to  be  equal  to 
such  as  are  known  on  the  other  side,  or,  which  is  the  same 
thing,  the  value  of  the  unknown  quantity  cannot  be  ascertain- 
ed till  we  transform  the  given  equation,  by  the  addition,  sub- 
traction, multiplication,  or  division  of  equal  quantities,  so  that 
we  may  fully  arrive  at  the  conclusion, 

a;=n, 
n  being  a  number,  or  a  formula,  which  indicates  the  opera- 
tions to  be  performed  upon  known  numbers.  This  number 
n  being  substituted  for  x  in  the  primitive  equation,  has  the  pro- 
perty of  rendering  the  first  member  equal  to  the  second.  And 
this  value  of  the  unknown  quantity,  as  has  been  already  ob- 
served, is  called  the.  root  of  the  equation,  this  word  has  not 
here  the  same  acceptation  as  in  (Art.  15.) 

169.  In  the  resolution  of  simple  equations,  involving  only 
one  unknown  quantity,  the  following  rules,  which  are  dedu- 
ced from  the  Articles  in  the  preceding  Section,  are  to  be  ob- 
served. 

RULE  I. 

When  the  unknown  quantity  is  only  connected  with  known  quan- 
tities hy  the  signs  plus  or  minus. 

170.  Transpose  the  known  quantities  to  one  side  of  the 
equation,  so  that  the  unknown  may  stand  by  itself  on  the 
other  ;  and  then  the  unknown  quantity  becomes  known. 

Ex.  1.  Given  a;+8  =  9,  to  find  the  value  of  a?. 
By  transposition,  a:=9  — 8, .-.  j;=l. 
Ex.  2.  Given  3a;— 4=2a;  +  5,  to  find  the  value  of  ap. 

By  transposition,  3a;— 2a;=5  +  4,  .*.  a;^9. 
Ex.  3.  Given  x\-a=ia-\-^^  to  find  the  value  oi x, 
,   By  taking  a  from  both  sides,  we  have 

a;=5  ;    or  by  transposition, 
a;=a— a+5  ;  but  a—a=iO  .'.  a?=5. 
10* 


102  SIMPLE  EQUATIONS. 

Ex.  4.  Given  9 — x=:2,  to  find  the  value  of  a?. 
By  changing  the  signs  of  all  the  terms,  we  have 
— 9  +  a:=— 2, 
by  transposition,  aci^Q— 2,  .•.x=7. 
It  may  be  remarked,  that  it  is  the  general  practice  of  Ana- 
lysts, to  make  the  unknown  quantity  appear  on  the  left-hand 
side   of  the   equation,   which  is  principally  the  reason  for 
changing  the  signs. 

Ex.  5.  Given  — h — x=a — c  to  find  x  in  terms  of  a,  b,  and  c. 
(161.  Cor.  1),  by  changing  the  signs  of  all  the  terms,  we 
have  5-f-a:  — c — a  ;  .'.  by  transposition,  a?z=c — b—a. 

Ex.  6.   Given  2x—4:-\-7  =  3x—2,  to  find  the  value  of  a;. 
(161.)  by  transposition,  2x—3x=4:—7—2,  and  (161.  Cor. 
1),  by  changing  the  signs,  3a; — 2a;=:74-2— 4  ;  but  3a; — 2a;= 
a:,  and  74-2 — 4=:5  ;  .-.  x=:5. 

Ex.  7.  Given  7a;-|-3— 5= 6a?— 2  +  7,  to  find  the  value  of  x. 

Ans.  a:  =  7. 
Ex.  8.   Given  3a;+5— 2— 2a?— 7=0,  to  find  the  value  of  x. 

Ans.  a?=4. 
Ex.  9.  Given  a?— 3-f4 — 6  =  0,  to  find  the  value  of  a?. 

Ans.  x==5. 
Ex.  10.  Given  7-|-a7ni2a?-f-12,  to  find  the  value  of  a?. 

Ans.  a;=— 5. 
Ex.  11.  Given  12  — 3a?=9— 2a?,  to  find  the  value  of  a?. 

Ans.  a?=:3. 

Ex.  12.   Given  x  —  a-i-b  —  c=:0,  to  find  the  value  of  a?  in 

terms  of  a,  b,  and  c.  Ans.  x  =  a  —  b-\-c. 

Ex.  13.  Given  x—a-\-b=2x — 2a+5,  to  find  the  value  of  x 

in  terms  of  a  and  b.  Ans.  x=a. 

Ex.  14.  Given  2x-{-a:=x-{-bj  to  find  a?  in  terms  of  a  and  b. 

Ans.  x=zb—a. 


RULE  n. 

171.  Transpose  the  known  quantities  to  one  side  of  the 
equation,  and  the  unknown  to  the  other,  as  in  the  last  Rule  ; 
then,  if  the  unknown  quantity  has  a  coefficient,  its  value  may 
be  found  by  dividing  each  side  of  the  equation  by  the  coeffi- 
cient, or  by  the  sum  of  the  coefficients. 

Ex.  1.  GiVen  3a?-f-9  =  18,  to  find  the  value  of  x. 

By  transposition,  3a?=18— 9,  or  3a?=:9  ;  dividing  both  sides 

3a;     9 
of  the  equation  by  3,  the  coefficient  of  a;,  we  have-^=-,  .*.  x 

o      o 

=3. 


SIMPLE   EQUATIONS.  103 

Ex.  2.  Given  2a— 3=i9— a;,  to  find  the  value  of  ar. 

By  transposition,  2a:+a?=i9+3, 
by  collecting  the  terms,  3a?=:12, 

1       1-   •  .        3a;     12 

by  division,  — =— ;  .*.  x=4. 

Ex.  3.  Given  7—4x=z3x—7,  to  find  the  value  of  «. 

By  transposition,  —4x  — 3a:  =  —  7 — 7, 

by  collecting  the  terms,  — 7£c=  — 14, 

by  changing  the  signs,  7a:=14, 

,      ,.  .  .        7x     14  - 

by  division,  -— =  — -  ;  /.  a;=2. 

Ex.  4.  Given  6a:+10  =  3a:+22,  to  find  the  value  of  ar. 

By  transposition,  6x — 3aj=:r22  — 10, 

by  collecting  the  terms,  3a: =12, 

u    A-      ■        3a;     12  . 

by  division,  — =—  ;  .-.  a:=4. 

o  o 

Ex.  5.  Given  aa?+^=--c  to  find  the  value  of  x  in  terms  of  a, 

b,  and  c. 

By  transposition,  ax=c—b, 

,      ,.  .  ,       ax     c — b  c—b 

by  division,  — = ;  .*.  x= . 

a         a  a 

The  value  of  x  is  equal  to  c— 5  divided  by  a,  which  may 
be  positive  or  negative,  according  as  c  is  greater  or  less  than 

9 5 

b\  thus,  if  c=r 9,  ^=5,  0=2,  then  a;=— -— =2  ;  if  c=12,  &= 

2 

16,  and  a=i2,  then,  — "^ — =^=—2. 

Ex.  6.  Given  3a;— 4  =  7a;— 16,  to  find  the  value  of  x. 

Ans.  a =3. 
Ex.  7.  Given  9— 2a;=3a;— 6,  to  find  the  value  of  ar. 

Ans.  a?=3. 
Ex.  8.  G'lYen' ax"^ -\-hxz=z9x'^ -\-cx,  to  find  the  value  of  x  in 

c— i 
terms  of  a,  b,  &c.  Ans.  x=z -. 

Ex.  9.  Given  a;— 9  =  4a;,  to  finj  the  value  of  a?.     • 

Ans.  x=. — 3. 

Ex.  10.  Given  5aa;— c=i— 3<za;,  to  find  the  value  of  x  in 

b-\-c 
terms  of  a.  b,  and  c.  Ans.  x=  —^ — . 

oa 

Ex.  11.  Given  3a;— 1-|-9— 5a;=0,  to  find  the  value  of  a;. 

Ans.  ic=4. 

Ex.  12.  Given  ax=ab—ac,  to  find  the  value  of  x. 

Ans.  xzz^b—c. 


104  SIMPLE   EQUATIONS. 

Ex.  13.  Given  x'^-\-2x=(x-\-a)'^,  to  find  the  value  of  x. 


Ans.  x=-- — -- 
2— 2a 


Ex.  14.  Given  (x — iy=:x-{-l,  to  find  the  value  of  x. 

Ans.  x=3. 

Ex.  15.  Given  a:3+2a;2+a;=:(a;24-3a:)x(a:— 1)4-16,  to  find 

the  value  of  x.  Ans.  x=4:. 


RULE    III. 

172.  If  in  the  equation  there  be  any  irreducible  fractions,  in 
which  the  unknown  quantity  is  concerned,  multiply  every  term 
of  the  equation  by  the  denominators  of  the  fractions  in  succes- 
sion, or  by  their  least  common  multiple  ;  and  then  proceed  ac- 
cording to  Rules  I,  and  II. 

2x 
Ex.  1.  Given f-l=a;— 9,  to  find  the  value  of  a;. 

4 

Multiplying  by  4,  2x4- 4= 4a;— 36, 
by  transposition,  2a;— 4a:  =  —  36 — 4, 
by  collecting  the  terms,  — 2x= — 40, 
by  changing  the  signs,  2a;=:40, 
2^     40 
by  division,  77=17- ;  •*.  a;=20. 

Ex.  2.  Given  -—-+3=5—-,  to  find  the  value  oix. 
^3  4 

2a;  2a; 

Multiplying  by  2,  x — — +6  =  10 — — , 

6x 

by  3,  3a;-2a;+18  =  30 — -, 

4 


....   by  4,    12a;— 8x+72  =  120-6a;. 

by  transposing,  and  collecting,   10a;=48, 

10a;     48 
by  division,  jQ=YQi  .:  x=z4^. 

Or,  it  is  more  concise  and  simple  to  multiply  the  equation  by 
the  least  common  multiple  oj"  the  denominators  ;  because,  then 
the  equation  is  reduced  to  its  lowest  terms  ;  thus, 

Multiplying  by  12,  the  least  common  multiple  of  2,  3,  and  4, 

we  have,  6a;— 4a;-+-36  =  60  — 3a;, 

by  transposition,  5a; =24, 

.....       5aj     24  ^. 

by  division, —=— ;  .'.x=4f. 
5        5 

Ex.  3.  G'uen  a?— tt— 1=-+^,  to  find  the  value  of  ap. 
o  5     o 


SIMPLE  EQUATIONS.  105 

Here  30  is  the  least  common  multiple  of  3,  5,  and  6  ; 

n-r  .  •  ,  •       ,      o^   o^       30a;     „^      30a?  ,  30a; 
Multiplymg  by  30,  30a; 30=—-+-—, 

o  O  O 

.-.  30a;— 10a;— 30  =  6a;+5j;, 

by  transposition,  9a;  =  30, 

^     ,.  .  .       9a;     30     10 

by  division,— =:ry=y;  .•  a;=3J. 

Ex.  4.  Given-— a=:-— 3,  to  find  the  value  of  a;.- 
4  5 

Here  20,  the  product  of  4  and  5,  being  their  least  common 

multiple, 

*,r  .  .  .   •       ,      c.r.  20a;     „„       20a;      ^^ 
Multiplying  by  20,— 20a=— 60, 

t:  0 

.-.  5a;— 20a=4a;— 60, 
by  transposition,  5a;— 4a;=20a— 60, 
/.  a;=20a— 60. 

Ex.  5.  Given  — r-=-r->  to  find  the  value  of  a;. 

5       5       5 

-_,.,.       ,      ^    5aa;     5bx     5x2a 
Multiplying  by  5,  — —=-—~^ 

.'.  ax — bx=2a, 
by  collecting  the  coefficients,  (a— i)a;=2a, 

.•.by  division,  x= r. 

•^  a—b 

Ex.  6.  Given 1 — --= — [-3,  to  find  the  value  of  a;. 

c         2        a 

Here  2ac,  the  product  of  2,  a,  and  c,  being  the  least  common 

multiple, 

Multiplying  by  2ac,  4a2a;+3a6ca;=10ca;4-6ac,    ♦ 

by  transposition,  and  collecting  the  coeflicients,  we  shall  have 

(Aa^-^-Sabc — 10c)a:=6ac, 

.-.  by  division,  xz= — -. 

•^  4a2+3aoc  — 10c 

Ex.  7.  Given  3a;— ^^^^^ ^="~t^ A'  *o  ^"^  ^^®  ^^' 

me  of  X. 

Multiplying  by  12,  the  least  common  multiple, 

we  have  36a;— 3a;+12— 48=:20a;-h56— 1, 
by  transposition,  36a;— 3a;— 20a;=56  — 1+48— 12, 
or  13a;  =  91, 
,      ,.  .  .        13a;     91 
by  division,  Y3-=j3  ;  •••  a?=7. 


106  SIMPLE  EQUATIONS. 

Ex.  8.  Given—— —=— -,  to  find  the  value  of  a:. 

Ans.  x=l. 

Ex.9.  Given  ^+^  =  16-^,  to  find  the  value 

ofjc.  Ans.  a; =13. 

Ex.  10.   Given  ^1^+^=20—^^^ — ,  to  find  the  value  of  sc. 

Ans.  x=z22\. 

XI X     19 X 

Ex.  11.  Given  x-\ — = — - — ,  to  find  the  value  of  x. 

o  2 

Ans.  xz=b. 

T«      ,r,     ^-        a?— 5  ,  ^       284 — X       ^    ,   , 

Ex.  12.   Given  — \-Qx= ,  to  find  the  value  of  x. 

4  5 

Ans.  a:r=9. 
Ex.  13.  Given  3x+^^^=5+li^^=:5I,  to  find  the  value 

of  x.  Ans.  a?=7. 

Qx 4  18 4ar 

Ex.  14.  Given  — 2= — \-x,  to  find  the  value  of 

o  o 

X.  Ans.  a;=4. 

T.      ,=     r.-         «^— 3     hx-^2      2aj— 9     a:— 1         ^    ^    , 
Ex.  15.   Given  — — = — -— ,  to  find  the 

o  o  ^  «J 

8T 
value  of  a?.  Ans.  x=z 


106  +  20— 6a* 
T^      ,^    r^-         ^—1     ^+3     2x+l      a;— 3        ^    ,    , 
Ex.  16.  Given  — i~=~T7 X~'*^  ^"^  ^^®  ^^" 

lue  of  a;.  Ans.  a;=:  —  9^. 

RULE    IV. 

173.  If  the  unknown  quantity  be  involved  in  a  proportion, 
the  proportion  must  be  converted  into  an  equation  (Art  165); 
and  then  proceed  to  resolve  this  equation  according  to  the 
foregoing  Rules. 

Ex.  1.  Given  3a; —2  ;  4  : :  5a;— 9  :  2,  to  find  the  value  of  at. 
Multiplying  extremes  and  means,  we  have 
2(3a;— 2)  =  4(5a;— 9), 
or  6a;— 4  =  20a;— 36, 
by  transposition,  6a?— 20a:  =—36 +4  ; 
or  — 14a;=— 32, 
by  changing  the  signs,  143*= 32, 


SIMPLE  EQUATIONS.  107 

by  division,  ^-= j^  ;  •*•  a?=2f . 

'     Ex.  2.  Given  3a  :  x  :  :  b-{-5  :  a:— 9,  to  find  the  value  of  «. 
Multiplying  extremes  and  means,  we  have 
3a  .  (x—9)=x  .  (&4-5), 
•        or  3ax—27a=bx-\-5x, 
by  transposition,  3ax—bx — 5ir=27a, 
collecting  the  coeff's,  (3a— b — 5)x=27a, 

/.  by  division,  x=- — -. 

3a — 0  —  o 

Ex.  3.  Given  — --  :  a;— 5  : :  7:  '•  -:,  ^o  find  the  value  of  x, 
4  3     4 

Multiplying  extremes  and  means,  we  have 

3     /x-5\     2     .       .. 

4 -(^  =  3  •("-")' 

3j;— 15     2a;-10 

°'-T6-=-F-' 

by  clearing  of  fractions,  9a?— 45  =  32a; — 160, 

by  transposition,  9a:  — 32a7=45  — 160, 

collecting  and  changing  signs,  23a:=115, 

.....        23a:     115  . 

by  division,  — =_-  ;  .-.  a:=5. 

Ex.  4.  Given  2a:— 3  :  a:— 1  :  :  4a:  :  2a:4-2,  to  find  the  value 
of  a:. 

Multiplying  extremes  and  means,  we  shall  have 
(2a:— 3)  .  (2a:+2)==4a:(a— 1), 
or  4a:2— 2a?— 6  =  4a:2— 4af, 
by  transposition,  &c.,  2a:=6, 
.-.  by  division,  a?=:3. 
Ex.  5.  Given  a-^x  :  b  :  :  c— a?  :  d,  to  find  the  value  of  x  in 
terms  of  a,  b,  c,  and  d. 

Multiplying  extremes  and  means,  ad-\-dx=bc — Ja;, 

by  transposition,  bx-\-dx  =  bc-^adj 

or  {b  +  d)x=bc-^ad, 

,      ,.  .  .  be— ad 

.'.  by  division,  x=   .  ,  -j-. 
o-\-a 

/p ]  ^ 

Ex.  6.  Given  — -—  :  a:+2  :  :  -  :.l,  to  find  the  value  of  a?. 
3  4 

Multiplying  extremes,  &c.,  — — — — , 

•  3  4 

clearing  of  fractions,  4a?— 4  =9a:4- 18, 
by  transposition,  4a?— 9a; =18+4, 


108  SIMPLE  EQUATIONS 

changing  the  signs,  &c.,  5a;=:— 22, 

22 
/.by  division,  x= =  --4f. 

Ex.  7.  Given  2a?— 1  :  x4-l  :  :  -r-  •  t>  ^o  fij^cl  the  value  of  a;. 

2     4 

•Ans.  ic=  — IJ. 
Ex.  8.  Given  x-\-3  :  a  :  :  b  :  -  to  find  the  value  of  x. 

X 


Ans.  a;: 


ab  —  1 


1  3x 

Ex.  9.  Given  -:—-::  5  :  2a:— 2,  to  find  the  value  of  a?. 

2  4 

Ans.  a?=  — j^. 

7    4  4 


4    3  2a: 1 

Ex.  10.  Given  -  :  - : :  a;— 1  :  — : — ,  to  find  the  value  of  x. 


Ans.  a;=lj-^. 
Ex.  11.  Given  — - —  :  — - —  ;  ;  6  :  3,  to  find  the  value  of  a?. 


Ans.  a?=3. 


3a  3a 

^  III.    EXAMPLES  IN  SIMPLE  EQUATIONS, 

Involving  only  one  unknown  Quantity. 

174.  It  is  necessary  to  observe  that  an  equation  express- 
ing but  a  relation  between  abstract  numbers  or  quantities,»may 
agree  with  many  questions  whose  enunciations  would  differ 
from  that  of  the  one  proposed  :  but  the  principles  of  the  reso-. 
lution  of  equations  being  independent  of  any  hypothesis  upon 
the  nature  and  magnitude  of  quantities  ;  it  follows,  therefore, 
that  the  value  of  the  unknown  quantity  substituted  in  the 
equation,  will  always  reduce  it  to  0  =  0,  although  it  may  not 
agree  with  the  particular  question.  This  is  what  will  hap- 
pen, when  the  value  of  the  unknown  quantity  shall  be  nega- 
tive ;  for  it  is  evident  that  when  a  concrete  question  is  the 
subject  of  inquiry,  it  is  not  a  negative  quantity  which  ought 
to  be  the  value  of  the  unknown,  or  which  could  satisfy  the 
question  in  the  direct  sense  of  the  enunciation. 

The  negative  root  can  only  verify  the  primitive  equation 
of  a  problem,  by  changing  in  it  the  sign  of  the  unknown  ;  this 
equation  will  therefore  agree  then  with  a  question  in  which 
the  relation  of  the  unknown  to  the  known  quantities  shall  be 
different  from  that  which  we  had  supposed  in  the  first  enuncia- 


SIMPLE   EQUATIONS.  109 

tion.  We  see  therefore  that  the  negative  roots  indicate  not  an 
absolute  impossibility,  but  only  relative  to  the  actual  enuncia- 
tion of  the  question. 

The  rules  of  Algebra,  therefore,  make  not  only  known  certain 
contradictions,  which  may  be  found  in  enunciations  of  problems  of 
the  first  degree ;  but  they  still  indicate  their  rectification,  in  ren- 
dering sub  tractive  certain  quantities  which  we  had  regarded  as 
additive,  or  additive  certain  quantities  which  we  had  regarded  as 
subtractive,  or  in  giving  for  the  unknown  quantities,  values  affect- 
ed with  the  sign  — . 

Hence,  it  follows,  that  we  may  regard  as  forming,  properly 
speaking,  but  one  question,  those  whose  enunciations  are  not 
connected  to  one  ^another  in  such  a  manner,  that  the  solution 
which  satisfies  one  of  the  enunciations,  can,  by  a  simple 
change  of  the  sign,  satisfy  the  other. 

We  must  nevertheless  observe  that  we  can  make  upon  the 
signs  and  values  of  the  terms  of  an  equation,  hypotheses  which 
do  not  agree  with  the  enunciation  of  a  concrete  question,  whereas 
the  change  which  we  will  make  in  this  enunciation  might  be 
always  represented  by  the  equation. 

These  principles,  which  will  be  illustrated  by  examples,  are 
applicable  to  equations  of  all  degrees,  and  to  determinate  equa- 
tions containing  many  unknown  quantities. 

The  question  which  conducts  to  the  equation, 
ax-^b^=cx-\-d, 
is  not  well  enunciated  for  a>c,  and  by-d,  since  the  first  mem- 
ber is  greater  than  the  second. 

Thus  the  formula 

d-b 

x= , 

a — c 

gives  for  x  a  negative  value  ;  but  by  rendering  the  unknown 
X  negative,  the  equation  is  changed  into  the  following, 

b^ax=:id—cx, 
which  is  possible  under  the  above  relations  between  a  and  c, 
b  and  d,  and  which  gives  then  for  x  an  absolute  value. 

If  we  have  b^d  and  c^a,  the  two  subtractions  become  im- 
possible in  the  formula 

d-b 

x=z  ; 

a — c 

but  in  order  to  resolve  the  equation,  let  us  subtract  cx-\-h 

from  both  members,  which  would  be  impossible,  because  that 

cx-\-b  is  greater  than  each  of  the  two  members :  we  must 

11 


110  SIMPLE  EQUATIONS. 

therefore,  on  the  contrary,  take  away  ax-\-d  from  both  sides, 
and  it  becomes 

b — d=.cx — ax  ; 
from  whence  we  deduce 

_b-d 
c~a 
This  formula  compared  to  the  preceding,  differs  from  it  in  this, 
that  tlte  signs  of  both  terms  of  the  fraction  are  changed. 

We  may  therefore  conclude,  that  we  can  operate  on  negative 
isolated  quantities,  as  we  would  do  if  they  had  been  positive. 

These  principles  will  be  clearly  elucidated,  when  we  come 
to  treat  of  the  solutions  of  Problems  producing  simple  Equa- 
tions :  we  shall  now  proceed  to  illustrate  the  Rules  in  the  pre- 
ceding Section,  by  a  variety  of  practical  examples. 

T?       in-         01.    3a:-ll      5a: -5     97 -Tar  _    ,    , 

Ex.  1.  Given  21-f  — r^ — =—x 1 — ,  to  find  the 

lo  o  Z 

value  of  X. 

Multiplying  both  sides  of  the  equation  by  16,  the  least  com- 
mon multiple  of  16,  8,  and  2,  we  shall  have 

336  +  3a:— ll  =  10j:-I0  +  776-56a7; 
/.  by  transposition, 

3a?- 10a?4-56ar=ll- 10  +  776—336, 
or  49a?=441  ; 

441 
by  division,  ^—-7^^  .'.x=9. 

^/p, 5  2a: 4 

Ex.  2.  Given  a; -I — =12 — ,  to  find  the  value  of  r. 

Multiplying  both  sides  of  the  equation  by  6,  the  product  of 

2  and  3,  which  is  the  least  common  multiple,  we  have 

6a:H-9a^-15  =  72— 4a:-f  8  ; 

/.by  transposition,  6a?-|-9a:H-4a7— 72-f  8+15,- 

or  19a:=95  ; 

95 
by  division,  a?=:— ,  .'.  x=b. 

2x 4 

In  this  example,  when  the  fraction — ,  is  multiplied 

o 

I2x 24 

by  6,  the  result  is =— (4a:— 8)=— 4a?-|-8,  or, 

which  is  the  same  thing,  when  the  sign  —  stands  before  a 
fraction,  it  may  be  transformed,  so  that  the  sign  -f-  may  stand 
before  it,  by  changing  the  sign  of  every  term  in  the  numerator ; 
therefore,  we  make  the  above  step  —4a; +8,  and  not  4a;— 8. 


SIMPLE  EQUATIONS.  lU 

Ex.  3.  Given  4x i" =«H i" — (-24,  to  find  the  value 

of  a?. 

Multiplying  by  10,  the  least  common  multiple,  and  we  have, 
40a;— 5a: -j-5  =  10a; -f4jc— 4+240, 
by  transposition,  40a; — 5a;— 10a; — 4a;=:240— 4 — 5, 
or,  40a;— 19a;=231  ; 

and  21a;=z231,  * 

by  division,  x=—-, .-.  a;=ll. 

Ex.  4.  Given  2a;----|-l=5a;— 2,  to  find  the  value  of  a?. 

Multiplying  by  2,  we  have, 

4a;— a;-f2  =  10a;— 4, 
.-.  by  transposition,  4«— a;— 10*=  —4—2, 

or  — 7x=— 6, 
by  changing  the  signs,  7a; =6, 

.-.by  division,  x=-. 

Ex.  5.  Given  ^ax—2bx=3b—'a,  to  find  the  value  of  a;. 

Here,  3aa;— 2fta;=(3a— 2^)a;,  by  collecting  the  coefficients 

of  X.     Therefore, 

{3a—2h)x—3b  —  a, 

.      ,.  .  .               3b  =  a 
by  division,  x= -r, 

Ex.  6,  Given  bx-{-xz=:2x-{-3a,  to  find  the  value  of  a;. 

by  transposition,  Z>a;-4-a:— 2a;=3a, 
or  (b  —  l)x=3ay 

.*.  by  division,  x=- — -. 

3a;  X  2x 

Ex.  7.  Given c+Y=4a;-| — -,  to  find  the  value  of  a;. 

a  0  a 

Multiplying  by  abd,  we  have, 

3bdx  —  abcd-\-  adx  =z  4abdx  -\-  2abXf 

by  transposition,  3bdx-\-adx—4abdx — 2abx  =  abcd, 

or  {3bd-{-ad—4:abd—2ab)x=abcd, 

.       ,.   .  .  abed 

•.  by  division,  x=—rr-, — j — 7-7-5 — ;l— ?• 
•^  3bd-\-ad—4:abd—2ab 

Ex.  8.  Given  c— 7^+7^  =  ^+<^j  to  find  the  value  of  x. 
5     6     6 

Multiplying  by  30,  the  product  of  5  and  6,  the  product  be- 
comes 


112  SIMPLE  EQUATIONS. 

6x--5x-{-5a=30b  +  30c; 

by  transposition,  6x—5x=30b-\-30c—5a, 

and .-.  x=z30b+30Q—5a. 

•«     «    r>.-        12  — a;    ^       144-a;      ,     «         ,.    -,   i         , 

Ex.  9.  Given — -— :  5x —  : :  1  :  8,  to  find  tne  value 

y  o 

of  a;. 

Multiplying  extremes  and  means,  we  have 

96—80;     ^        l4:  +  x 

9  3     ' 

Multiplying  by  9,  the  least  common  multiple, 

96— 8a:=45a:— 42  — 3ar, 

by  transposition,  — 45a7 — 8a;-f  3a;= — 96 — 42, 

by  changing  the  signs,  45a!+ 8a;— 303=96+42, 

or  50o;=138, 

^     ,.  .  .  138     ^19      • 

.-.by  division,  a;=--=2-. 

in      ,^^'        0.x — h  ,  a     bx      bx—a  ^    ,     ,  . 

Ex.  10.    Given  — ro~~^ o — »  ^^   """   ^^®  value 

of  X. 

Multiplying  by  12,  the  least  common  multiple  of  the  deno- 
minators, and  the  equation  will  become, 

3ax—3b-{-4a=z6bx—4bx+4a,     ,    .     (1), 
by  taking  away  4a  from  each  member,  we  shall  have 

3ax—3b=z6bx—4bx=2bxy 
by  transposing  —3^  and  2bx,  it  becomes 

3ax—2bx=3b, 
by  collecting  the  coefiicients  of  x,  we  shall  have 

{3a—2b)x=3b, 

by  division,  a.=3-?^. 

Ex.  11 .  Given  2ax-\-b=3cx-\-4a,  to  find  the  value  of  x. 

by  transposition,  2ax—3cx  =  4a — by 
by  collecting  the  coefficients,  (2a— 3c)a:=4a  — i, 

.*.  by  division,  a:= —. 

^  '        2a  — 3c 

Ex.  12.  Given  19a?+ 13=59— 4ar,  to  find  the  value  of  x. 

by  transposition,  1 9a; -f- 4a; =59  — 13, 
or,  23a;  =  46 ; 
.-  by  division,  a?=2. 

X 

Ex.  13.   Given  3a;+4—  -=46--2a:,  to  find  the  value  of  op. 

Multiplying  both  sides  by  3, 

9a;4-12— a;=138— 6a?, 


SIMPLE  EQUATIONS.  113 

by  transposition,  9x-\-6x'—x=\38—'l2, 
or  I4a;=126  ; 

by  division,  a;=-— -,  .*.  a?=9. 

Ex.  14.  Given  a;2_|_i5a._35a:_3a;2^  to  find  the  value  of  x. 
Dividing  every  term  by  x, 

074-15 =35— 3a:, 
by  transposition,  ar-f3x=35  — 15, 
or  4a;=20  ; 
.•.x=z5. 

Ex.  15.  Given-— j+ 10=^— ^4- 11,  to  find  the  value  of  a?. 

Here  12  is  the  least  common  multiple  of  6,  4,  3,  and  2  ; 

.*.  multiplying  both  sides  of  the  equation  by  12, 

2aj— 3a;+120=4a;— 6a;+132; 

by  transposition,  2a?— 3aj— 4a? + 6a;  =132  — 120, 

or  8j?— 7a:=12  ; 

.-.  a?=12. 

f-.      -«    ^-        a?— 1  ,  23 — A     _      4  +  a;        -    ,   ,  t 

Ex.  16.  Given — -— + — - — -=7 — ,  to  find  the  value 

7  5  4 

of  X.  Ans.  x=8. 

■r.      ,^    r.'        7a:4-5        164-4a:  ,  ^     3a?+9  ^    ,     , 

Ex.  17.  Given — ^ ^ \-6=  ,  to  find  the 

value  of  a:.  ,  Ans.  a?=l. 

^       ,o    r^-        17-3a;       4a;4-2      ^      .        7a;H-14^    ^    , 

Ex.  18.  Given ■ — =5— 6a:H j- — ,tofind 

5  3  o 

the  value  of  x.  Ans.  a? =4. 

17      ion-  3a?-3  ,  ,       20-«       6a;-8     4a:-4 

Ex.  19.  Given  x h— +4  =  —^ ^ ^—r~» 

5  2  7  5 

to  find  the  value  of  x.  Ans.  a; =6. 

T^      «^    r.-        4a:— 21  .  „,  .  57— 3a;     „,,       5a;  —  96 
Ex.  20.  Given       ^      +3j+ ^^"=241 12~  "" 

11a:,  to  find  the  value  of  a:.  Ans.  a:=21. 

17      o,     n-        6a:+18      ^,       ll-3a:     ^         ._       13-a: 
Ex.  21.  Given— j^ 4f _-=5a;-48 ^^ 

— — ,  to  find  the  value  of  a?.  Ans.  a:  =10. 

a^—Sbx  ,  ,  ebx—5a^ 

Ex.  22.  Given  ax ah^  =z  bx  -] 

a  ^ict 

5a;+4a        .    .  v.        ,        r  a  4g&'^-10a 

— - — ,  to  find  the  value  of  x.  Ans.  a;=    .   __g,    . 

11* 


114  SIMPLE  EQUATIONS. 

Ex.  23.  Given  -^^^ _Z__^_^  to  find  the  value  of  ar. 

Ans.  a;=8. 
Ex.  24.  Given -^-f-^-___=:_^,  to  find  the  value 

^^^-  Ans.  a:==4. 

T?^    OK    o-        4a;+3  ,  7ir— 29     8a:4-19 
Ex.  25.  Given—^— +— -__:=_J^,to  find  the  value 

^^^'  Ans.  a:=6. 

Ex.  26.  Given  12-a;  :  ^  : :  4  :  1,  to  find  the  value  o{  x. 

Ans.  a;=:4. 
Ex.  27.  Given  -^±i  :  i^Zlf  : :  7  :  4,  to  find  the   value 

of  a;.  Ans.  a:=2. 

Ex.  28.  Given  (2a;+8)2=4a;2+l4a:+172,  to  find  the  value 

o^^-  Ans.  a  =  6. 

Ex.  29.  Given— ^—+20?=—.^+ 16,  to  find  the  value 
°^^-  Ans.  a; =7. 

Ex.  30.  Given  I^+4  =  ?i^ii+?f!+L^  to  find  the  va- 
^"6  0^^-  .  Ans.  a:=::3. 

2  "^2~~2 


Ex.  31.  Given  ^+^r=r-^-,  to  find  the  value  of  a;. 


.  1 

Ans.  a:= 


■3a— 1 

Ex.  32.  Given  2a:--^  +  15=i?^,tofindthevalue 
r  d  5 

°*^-  Ans.  a:=12. 

Ex.  33.  Given   5aa;--26  +  4^a:=2a;+5c,  to  find  the  value 

of^-  ■  Ans.  cc=-  ^'^^^    . 

5a+46  — 2 

T?^    -iA     n         2a:  — 5  ,  19  — x     lOar— 7       5  ^    ,    , 

tiX.  34.  Criven = _  to   find  the 

18     ^     3  9  2' 

value  of  a;.  Ans.  a; =7. 

Ex.  35.  Given  x-  ?^±l=^±i,  to  find  the  value  of  x. 
3  4 

Ans.  a;=13. 

Ex.  36.  Given  -^ l±fl^3g^5^^  to  find  the  value 

<>f«-  Ans.  a:=9. 


SIMPLE  EQUATIONS.    ,  /^^ 


115 


Ex.  37.  Given4a;-^l5±5f^l5. 


lue  of  X. 

Ex.  38.  Given 
value  of  X: 


21— 3a?      4a;+6 


=6— 


7a;  4- 11 

4 

5a;+l 


Ex.  39.  Given  7?+^- 
o         4 


6     ~  4 

7a:+3     8a;+19 


,  to  find  the  va- 

Ans.  07=3. 

,  to   find  the 

Ans.  a;=3. 

,  to  find  the 


value  of  X. 

Ex.  40.  Given 
the  value  of  x. 

Ex.  41.  Given  a; 


6a; -f  8 


16  8 

Ans.  a;=7. 

5a;+3     27— 4a;       3a;+9         ^    , 
^  -  — ^,  to  find 


11  2  3 

27-9a;      5a;4-2     61 


Ans.  a;=6. 
2a;+5     29+4a; 


11 
5a:-l 


to  find  the  value  of  x. 

Ex.  42.  Given 
value  of  X. 

Ex.  43.  Given 
of  a;. 

Ex.  44.  Given 
of  a;. 

Ex.  45.  Given 
the  value  of  x. 


6 

15a;+8_ 


12 


13 
7a;— 2 


:3a;- 


31 


12     ' 
Ans.  a:=5. 

to  find  the 

Ans.  a;=:9. 


2 

10+a; 
5 

17-4a; 


Ex.  46.  Given  16a;+5 
the  value  of  a;. 


10 

4a;— 9 
7       *  * 

15+2a; 
3 

4a;-f  14 
9a;  4-31 


Ex.  47.  Given 
the  value  of  x. 


4a;  +  3 
6a;— 43 


=6f  —  -,  to  find  the  value 

Ans.  a;=3. 

:  :   14  :  5,  to  find  the  value 

Ans.  a;=4. 

2a;  :  ;  5  :  4,  to   find 

Ans.  a;=3. 

:   36a;-f  10  :  1,  to  find 

Ans.  x=z5. 

2a;+19  :    3a;— 19,  to  find 

Ans.  a;=8. 


7a;_L9  lOa;^ 18 

Ex.  48.  Given  5a:4-^   7^  =  9+  - — r-r:—,  to  find  the  va- 


lue of  X. 

Ex.  49.  Given 

Ex.  50.  Given 
lue  of  X, 


4a;+3 
9a;4-20     4a; 


2a;+3 


Ans.  a;=3. 


36 


12     X 

--4-tj  to  find  the  value  of  a;. 
5x — 4      4 

Ans.  a;=8. 


20a;4-36  ,  5a; 4- 20     4a;  ,  86 


25 


9a;- 16      5    '  25 


\-—-j  to  find  the  va- 


Ans.  a;=4. 


116  SIMPLE  EQUATIONS. 

Ex.  51.  Given — ;= — - — ,  to  find  the 

18  13a:— 16         9     ' 

value  of  a;.  Ans.  a;=::4. 

Ex.  52.  Given — -r- f--^ — rTT^ — t-a — ,  to  find  the  va- 

28  6a?+14  14 

lue  of  a?.  Ans.  x=z7. 

Ex.  53.  Given -^^-T =ac+-^,  to  find  the  value  of  a?. 

ox  b 

Ans.  «=-. 
c 

Ex.  54.  Given r— = ^-,  to  find  the  value  of  x. 

a-{-ox      e+Jx 

.  ad — ce 

Ex.  55.  Given  -; — \-- — f-^ — f--r-=^j  to  find  the  value  of  x. 
ox     ax    jx      hx 

A««   ^_adfh-\-bcfk+bdeh+  bdfg 
Ans.  X-  -^^  . 


find 
the  value  of  x.  Ans.  x= 


Ex.  56.  Given  (b-^x).(b-{-x)—a.{b+c)z=—--^x'^,   to 

ac 

T' 

T?       :.^    n-        3a;-3     3a:-4      _.      27+40;  .    .    ,. 

Ex.  57.  Given — — =5J — ,  to   find  the 

value  of  a;.  Ans.  a; =9. 

T-       .o    r.-        4a;-34     258-5a:     69-a;  ,     .    ,    . 

Ex.  58.  Given  — — = — - — ,  to  find  the  va- 

lue  of  a;.  Ans.  a;=51. 

4a:-2     2a;+ll      7— 8a;  ^    ,     , 

Ex.  59.   Given  2a; tT~= — \ ^~"'  ^^  ^"^   ^^® 

value  of  a;.  Ans.  a; =7. 

^      .^    r.'        2a;4-l      402-3a;     _      471 -6a;  ^    ... 
Ex.  60.  Given —-^; — — =9 ,  to  find  the 

value  of  X.  Ans.  a;=:72. 

3<z4-a;  6 

Ex.  61.  Given 5=-,  to  find  the  value  of  a;. 

X  X 

A  3a— 6 

Ans.  a;= — : — . 


117 

CHAPTER  IV. 

ON 

THE  SOLUTION  OF  PROBLEMS, 

PRODUCING  SIMPLE  EaUATIONS. 

175.  The  solution  of  a  problem  is  the  method  of  discover- 
ing, by  analysis,  quantities  which  will  answer  its  several  con- 
ditions ;  for  this  purpose,  there  are  four  things  to  be  distin- 
guished : 

I.  The  given,  that  is  to  say,  the  known  quantities,  enunci- 
ated in  the  problem,  and  the  quantities  that  are  to  be  found. 

II.  The  translation  of  the  problem  into  algebraic  language, 
which  is  composed  of  the  translation  of  every  distinct  condi- 
tion that  it  contains  into  an  algebraic  equation. 

III.  The  resolution  of  the  equations,  that  is,  the  series  of 
transformations  which  the  immediate  translation  must  under- 
go, in  order  to  arrive  at  an  equation  containing  in  the  first 
member  one  unknown  quantity  alone  in  its  simple  state,  and 
in  the  other  a  formula  of  operations  to  be  performed  upon  the 
representations  of  given  numbers. 

IV.  Finally,  the  numerical  valuation,  or  the  geometrical 
construction  of  this  formula. 

176.  Algebraic  problems  and  their  solutions  may  be  con- 
sidered as  of  two  kinds,  that  is,  numerical  and  literal,  or  par- 
ticular and  general.  In  the  numerical,  or  particular  method 
of  solution,  unknown  quantities  are  represented  by  letters,  and 
the  known  ones  by  numbers,  as  in  arithmetic.  In  the  literal, 
or  general  solution,  all  quantities,  known  and  unknown,  are 
represented  by  letters,  and  the  answers  given  in  general  terms. 
A  problem  solved  in  this  way,  furnishes  a  theorem,  which  may 
be  applied  to  the  solution  of  all  questions  of  the  same  kind. 

^I.    SOLUTION  OF  PROBLEMS  PRODUCING  SIMPLE  EQUATIONS, 

Involving  only  one  unknown  Quantity. 

177.  If  from  certain  quantities  which  are  known,  another 
quantity  be  required  which  has  a  given  relation  to  them,  let 
the  unknown  quantity  be  represented  by  x  ;  then,  the  condi- 
tion enunciated  in  the  problem  being  clearly  understood,  it  can 
be  easily  translated  into  an  algebraic  equation,  by  means  of  the 


118  SOLUTION  OF  PROBLEMS. 

signs  pointed  out  in  the  Introduction.  Having  now  brought 
the  question  into  an  algebraic  form,  the  value  of  the  unknown 
quantity  can  be  readily  found  by  the  application  of  the  rules 
delivered  Chap.  III. 

Or,  if  there  be  more  than  one  unknown  quantity  required, 
and  that  they  bear  given  relations  to  one  another,  instead  of 
assuming  a  symbol  to  represent  each  of  them,  it  is  more  con- 
venient to  assume  one  only,  and  from  the  conditions  of  the 
problem  to  deduce  expressions  for  the  others  in  terms  of  that 
one  and  known  quantities.  And  as  the  number  of  conditions 
ought  to  be  one  more  than  the  number  of  quantities  thus  ex- 
pressed, there  will  remain  one  to  be  translated  into  an  equa- 
tion ;  from  which  the  value  of  the  unknown  quantity  may  be 
determined  as  above  ;  and  this  being  substituted  in  the  other 
expressions,  their  values  also  may  be  discovered. 

Problem  I. 

What  number  is  that,  to  which  17  being  added,  the  sum 
will  be  48  ? 

Let  the  required  number  be  represented  by  x  : 
Then  by  the  problem,  a;4-17  =  48  ; 

by  transposition,  a;=:48  — 17  : 
.-.  a;=31. 

Prob.  2.  What  number  is  that,  from  which  a  being  sub- 
tracted, the  remainder  is  b  1 

Let  X  represent  the  number  required. 

Then  by  the  problem  x—a=:b; 

by  transposition,  x=a-\-b. 

Here,  if  05=16,  and  ^>  =  14  ;  then  a:=16-f  14  =  30  ;  that  is, 
30  is  a  number,  from  which  16  being  subtracted,  the  remain- 
der is  14. 

Prob.  3.  To  find  a  number  which,  being  subtracted  from 
a,  leaves  b  for  a  remainder. 

Designating  the  unknown  number  by  x,  we  shall  have  this 
translation, 

a — a?=6,  .'.  x=a — b. 

178.  If  we  suppose  a=  10,  5  =  4,  we  shall  have  a?  =  6  ;  then 
the  subtraction  is  arithmetically  performed.  But  if  we  had 
a=10,  5  =  14,  we  must  subtract  14  from  10,  which  cannot  be 
done  except  in  part,  or  that  with  respect  to  the  portion  of  14 
equal  to  10. 

The  excess,  in  as  much  as  it  exists  subtractively,  will  indi- 
cate that  the  number  x  of  which  it  is  the  representation  must 


PRODUCma  SIMPLE  EQUATIONS.  119 

enter  negatively  in  the  enunciation  where  it  is  already  sub- 
tracted from  the  number  a,  so  that  the  enunciation  of  the  pro- 
blem is  corrected  and  brought  to  these  terms  :  to  find  a  num.' 
her  which  being  added  to  10,  the  sum  will  be  14;  a  problem 
whose  translation  is,  designating  the  unknown  quantity  by  a:, 

10  +  a:=:14;    .-.  a;=14  — 10=4  ; 
whereas,  the  translation  in  the  former  case  would  be 
10  — a;=14;   .-.aj^lO  — 14,  or  £c=--4. 

The  negative  root  —4,  satisfies  the  equation  of  the  problem, 
besides  it  announces  a  rectification  in  the  enunciation  ;  this  is 
what  appears  evident,  since  the  subtraction  of  a  negative 
quantity  is  equivalent  to  the  addition  of  a  positive,  (Art.  63). 
In  fact,  as  has  been  already  observed,  (Art.  174),  it  makes 
known  that  the  enunciation  ought  to  be  taken  in  an  opposite 
sense  to  that  which  we  first  proposed  in  the  problem. 

Prob.  4.  A  person  lends  at  interest  for  one  year  a  certain 
capital  at  5  per  cent ;  at  the  end  of  the  year,  according  to 
agreement,  he  is  to  receive  a  sum  b,  besides  the  principal  and 
interest,  and  the  whole  sum  he  receives  must  be  equal  to  the 
capital.     I  demand  what  is  the  capital  ? 

Let  the  capital  be  designated  by  x  : 

Since  100  dollars  becomes  at  the  end  of  the  year  105  dollars, 

we  shall  have  the  capital  at  the  same  time  by  this  proportion, 

105a; 
100  :  105  : :  a;  :  -r^=  the  capital. 

105a; 
The  sum  +&,  by  the  problem,  must  be  equal  to  x,  we 

have  therefore  the  equation 

105;r 

-— +Z>=a;;  .-.  105a;+ 1006  =  100a; ; 

by  transposition,  5x=  —  l00h; 
.;.  by  division,  x=—20b. 
179.  Thus  the  capital  shall  be  — 20b.     This  answer  does 
not  agree  with  the  problem,  and  still  if  this  value  —206,  be 
substituted  for  x  in  the  equation  found,  we  obtain 

and,  performing  the  operations  indicated  in  the  first  member,  it 
becomes 

—20b=—20b, 
which  is  true.     This  value  of  x,  although  it  is  negative,  satis- 
fies the  equation  of  the  problem,  as  has  been  already  observed 
(Art.  174),  since  its  two  members  become  identicalli/ eqaal  by 
making  the  proper  substitution . 


120  SOLUTION  OF  PROBLEMS 

If  we  return  again  to  the  enunciation,  we  discover  that  it  ia 
impossible  that  a  capital  augmented  by  the  interest  would  re- 
main equal  to  itself,  and  that  much  more  this  impossibility 
takes  place,  if,  besides  the  interest,  we  add  to  it  a  sum  h  ;  it 
is  necessary  therefore  that  one  of  these  two  parts,  namely, 
the  interest  at  5  per  cent,  and  h,  be  subtracted. 

In  fact,  if  we  carry  into  the  first  equation  this  circumstance 
— X,  which  is  but  x=z  —  a  number,  we  find 

105     ,  ,  105a;      ^ 

-Ioo^+^=~^'-"--io^~^=^' 

a  translation  of  the  enunciation,  by  supposing  the  interest  ad- 
ditive to  the  capital,  in  which  case,  the  sum  b  ought  to  be 
subtracted. 

This  equation,  treated  as  the  preceding,  shall  give 
x  =  20b, 

If  the  interest  at  5  per  cent  be  subtracted  from  100,  in  which 

case  100  reduces  itself  to  95,  we  have  the  capital  x  at  the 

end  of  the  year,  by  the  proportion 

95a; 
100  :  95  :  :  a;  :  j7r^=  the  capital, 

consequently,  T7)(\'^  ' 

multiplying  by  100,  and  transposing,  we  shall 'have 
100^>=:5a:,  .-.  x=2Qb. 
The  negative  isolated  result,  that  is,  the  negative  value  of 
X,  would  announce  a  rectification  or  a  correction  in  the  terms 
of  the  enunciation,  and  the  problem  proposed  could  be  re-es- 
tablished in  two  ways. 

Prob.  5.  What  number  is  that,  the  double  of  which  exceeds 
its  half  by  6  ? 

Let  a;=  the  number  ; 

Then  by  the  problem,  2x—  -=6, 

.-.multiplying  by  2,  4a:— a;  =  12, 
or3a:  =  12, 
.'.  by  division,  a;  =  4. 
Prob.  6.  From  two  towns  which  are  187  miles  distant,  two 
travellers  set  out  at  the  same  time,  with  an  intention  of  meet- 
ing.    One  of  them  goes  8  miles,  and  the  other  9  miles  a  day 
In  how  many  days  will  they  meet  ? 

Let  x—  the  number  of  days  required  ; 
then  8a:  =  the  number  of  miles  one  travelled, 
and  9a:  =  the  number  the  other  travelled ; 


PRODUCING  SIMPLE  EQUATIONS.         121 

and  since  they  meet,  they  must  have  travelled  together  the 
whole  distance, 

consequently,  8a;4-  9a:=  1 87, 

or  17a?=187, 

.-.  by  division,  a; = 1 1 . 

Prob.  7.  What  number  is  that,  from  v^rhich  6  being  sub- 
tracted, and  the  remainder  multiplied  by  11,  the  product  will 
be  121  ? 

Let  x=  the  number  required  ; 

Then  by  the  problem  (a;— 6)  X  11  =  121, 

by  transposition,  lla;=121  +  66, 

or  lljr=187, 

.'.by  division,  a:  =  17. 

Prob.  8.  A  Gentleman  meeting  4  poor  persons,  distributed 
five  shillings  amongst  them  :  to  the  second  he  gave  twice,  the 
third  thrice,  and  to  the  fourth  four  times  as  much  as  to  the  first. 
What  did  he  give  to  each  ? 

Let  a?=  the  pence  he  gave  to  the  first, 

.■.2x=:  the  pence  given  to  the  second, 

and  3a;  =: to  the  third, 

4t= to  the  fourth. 

.-.by  the  problem,  a;-|-2a:-f  3ir+4a;=5  X  12  =  60, 

or  10a; --60, 

by  division,  x=zQ^ 

and  therefore  he  gave  6, 1 2, 18, 24  pence  respectively  to  them. 

Prob.  9.  A  Bookseller  sold  10  books  at  a  certain  price  ;  and 
afterwards  15  more  at  the  same  rate.     Now  at  the  latter  time 
he  received  35  shillings  more  than  at  the  former.     What  did 
he  receive  for  each  book  ? 
Let  a;=r  the  price  of  a  bool^. 

then  10a;=  the  price  of  the  first  set, 
and  15a;=  the  price  of  the  second  set 
,  but  by  the  problem,  15a;  =  10a;+35 

.-.by  transposition,  5a;=:35 
and  by  division,  a;=:7. 

Prob.  10.  A  Gentleman  dying  bequeathed  a  legacy  of  1400 
dollars  to  three  servants.  A  was  to  have  twice  as  much  as 
B  ;  and  B  three  times  as 'much  as  C.  What  were  their  re- 
spective shares  ? 

Let  a;=zG's  share, 

.-.  3a;=:B's  share, 

and.  6a; =A's  share  • 

i2r 


122  SOLUTION  OF  PROBLEMS. 

them  by  the  problem,  a^-f  3x+6a;=1400, 
or  10a:  =1400, 
.'.by  division,  a;=:140r=:C's  share. 
.-.  A  received  840  dollars  ;  B-,  420  dollars  ;  and  C,  140  dol- 
lars. 

Prob.  11.  There  are  two  numbers  whose  difference  is  15, 
and  their  sum  59.     What  are  the  numbers  1 

As  their  difference  is  15,  it  is  evident  that  the  greater  num- 
ber must  exceed  the  lesser  by  15. 

Let,  therefore,  x=  the  lesser  number; 
then  will  a?+15=  the  greater  ; 

.'.by  the  problem,  a;+a:4-15=:59, 

or  2a:-i-15=z59, 

by  transposition,  2a?= 59  —  1 5  =:  44, 

.'.by  division,  a:=:22  the  lesser  number, 

and  a:  4- 15=22  4- 15  =  37  the  greater. 

Prob.  12.  What  two  numbers  are  those  whose  difference 
is  9  ;  and  if  three  times  the  greater  be  added  to  five  times  the 
lesser,  the  sum  shall  be  35  ? 

Let  x=  the  lesser  number  ; 
then  a:+9=  the  greater  number. 
And  3  times  the  greater  =:3(a?+9)  =  3a;+27, 
5  times  the  lesser  =5a:. 

.-.  by  the  problem,  (3a;-f27)-f-5a?=35  ; 

by  transposition,  3a?  +  5a:=35— 27, 

or8a:=8; 

.•.by  division,  x=l  the  lessernumher, 

and  a:+9= 1  +  9  =  10  the  greater  number. 

Prob.  13.  What  number  is  that,  to  which  10  being  added, 
|ths  of  the  sum  will  be  66  ? 

Let  x=  the  number  required  ; 
then  a;  4- 10  =  the  number,  with  10  added  to  it. 

Now  fths  of  (.+  10)=f(.+  10)=?(^>=^^. 

But,  by  the  problem,  fths  of  (a;+10)=66  ; 

3a:+30 
.-.-^—=66; 

by  mukiplication,  3x4-30  =  330; 

by  transposition,  3a;=300; 

.-.  by  division,  a:=  1 00. 

Prob.  14.  What  number  is  that,  which  being  multiplied  by 


PRODUCING  SIMPLE  EQUATIONS.         123 

6,  the  product  increased  by  18,  and  that  sum  divided  by  9,  the 
product  shall  be  20  ? 

Let  x:=  the  number  required  ; 
then  6x=:  the  number  multiplied  by  6  ; 

6j;+18=  the  product  increased  by  18  ; 

and  — - — =  that  sum  divided  by  9, 
y 

1.      1  1.1         6rr+18     ^^ 

.'.  by  the  problem,  — - — =20 

by  multiplication,  6x4-18=20x9 
by  transposition,  6a?=180  — 18 
or6a;=162 
/,  by  division,  a;=27. 

Prob.  15.  a  post  is  Jth  in  the  earth,  fths  in  the  water,  and 
13  feet  out  of  the  water.     What  is  the  length  of  the  post  ? 

Let  x=^  the  length  of  the  post ; 
then  -=  the  part  of  it  in  the  earth, 

0 

3a: 

-— -=  the  part  of  it  in  the  water, 

and  13=  the  part  of  it  out  of  the  water. 

But  by  the  problem,  part  in  the  earth  +  part  in  water  + 
part  out  of  water  =  whole  part ; 

y(I)+(T)+-=- 

and  ^  X35  +  ^X35+I3x35  =  35a;; 
5  7 

or  7a: 4- 15a: +455  =  3 5a:; 

by  transposition,  455  =  35a: — 7a:— 15a: = 13a:, 

or  13a:=455  ; 

.*.  by  division,  a:=35,  length  of  the  post. 

Prob.  16.  After  paying  away  Jth  and  ith  of  my  money,  I 
had  850  dollars  left.     What  money  had  I  at  first  ? 
Let  X—  the  money  in  my  purse  at  first ; 

then-4--=  money  paid  away. 

But  money  at  first  —  money  paid  away  =  money  remaining ; 


.-.  by  the  problem  ^—(7+7)  =850> 


era:— -—-=850. 

4     7 


124  SOLUTION  OF  PROBLEMS 

Multiplying  by  28,  the  product  of  4  and  7 ,  which  is  the 
least  common  multiple, 

and28a:— I X  28— 1x28  =  850x28, 

or  28j;— 7a;— 4a?=:23800, 
.-.  17a;=:23800  ;  and  by  division,  a;=1400  dollars. 

Prob.  17.  What  number  is  that,  whose  one  half  and  one 
third,  plus  12,  shall  be  equal  to  itself? 

Let  x=  the  number  required  ; 

then,  by  the  problem,  a!=-+-+12  ; 

Now  to  clear  this  of  fractions,  multiply  by  6, 

and  6x=3x+2x-\-72  ; 

by  transposition,  60?— 50?= 72  ; 

/.a;=72. 

It  can  be  readily  proved  that  72  is  the  number  required  ; 

72     72 
thus,  — +— -|-12  =  36  +  24  +  12=72. 
2        o 

All  other  problems  in  this  Section  may  be  proved  in  like 

manner. 

Prob.  18.  To  find  a  number,  whose  half,  minus  6,  shall  be 
equal  to  its  third  part,  plus  10. 

Let  3?=:  the  number  required  ; 
then  by  the  problem,  -—6  =  -+ 10, 

.-.  clearing  of  fractions,  Sa?  — 36=2ji:  +  60, 

by  transposition,  3a; — 2a:=60-(-36, 

.-.  a;=96. 

Prob.  19.  Two  persons,  A  and  B,  set  out  from  one  place, 
and  both  go  the  same  road,  but  A  goes  a  hours  before  B,  and 
travels  n  miles  an  hour  ;  B  follows,  and  travels  m  miles  an 
hour.  In  how  many  hours,  and  in  how  many  miles  travel, 
will  B  overtake  A  ? 

Let  a;=  the  hours  that  B  travelled  ; 
then  x-{-a=  the  hours  that  A  travelled. 

Also  mx=z  the  nusnber  of  miles  travelled  by  B  ; 
and  n(x-\-a)=nx-\-na=  the  miles  travelled  by  A  ; 

.-.  by  the  problem,  mx=nx-^na  ; 

by  transposition,  mx—nx=.nay 

or  {m—'n)x—na ; 


PRODUCING  SIMPLE  EQUATIONS.         125 

,.  .  .        (m~n)x        na 

.'.  by  division,  -^ —= , 

m—n       m — n 

net 

.'.  x= ,  the  hours  that  B  travelled. 

m — n 

^,  .  na      ,         na-\-ma — na         ma       .       . 

Then  x-\-a  = f-<3f= =  ,  the   hours 

m — n  m — n  m — n 

that  A  travelled  ;  and  mx= =  the  miles  travelled. 

m — n 

180.  This  is  a  general  or  literal  solution,  because  m,  n,  a, 
may  be  any  numbers  or  quantities  taken  at  pleasure  ;  for  ex- 
ample, 

Let  a=9,  n=5,  and  m=:7  ; 

Then,  A  travels  9  hours  at  the  rate  of  5  miles  an  hour,  be- 
fore B  sets  out ;  and  B  follows  after  at  the  rate  of  7  miles  an 
hour. 

Now,  by  putting  these  values  of  a,  n,  and  m,  in  the  formula 

found  above  ;  we  have, 

na       9x5     45     „^-     ,     ,  ,       _,  ,,    , 

x= =- — -= — =22^,  the  hours  that  B  travelled  ; 

m—n     7—5      2  ^ 

,  ma       9X7     63     ^,,     ,     ,  ,,   j  ,       * 

and  x= =- — -=-—=311,  the  hours  travelled  by  A. 

m—n     7—5      2  ^  -^ 

And  ma;= 7x22^=  157^,  the  miles  travelled  by  each. 

pROB.  20.  Four  merchants  entered  into  a  speculation,  for 
which  they  subscribed  4755  dollars  ;  of  which  B  paid  three 
times  as  much  as  A  ;  C  paid  as  much  as  A  and  B  ;  and  D 
paid  as  much  as  C  and  B.     What  did  each  pay  ? 

Here,  if  we  knew  how  much  A  paid,  the  sum  paid  by  each 
of  the  rest  could  be  easily  ascertained  ; 

Let,  therefore,  «=  number  of  dollars  A  paid  ; 
3ir=  number  B  paid  ; 
4a;=  number  C  paid  ; 
and  7a;  =  number  D  paid  ; 

.-.  (a;+3a;+4a;+7a;=)15a:=4755, 
and  a?=-317. 
.-.they  contributed  317,  951,  1268,  and  2219   dollars  re- 
spectively. 

Prob.  21.  Let  it  be  required  to  divide  890  dollars  between 
three  persons,  in  such  a  manner,  that  the  first  may  have  180 
more  than  the  second,  and  the  second  115  more  than  the  third. 

Here,  it  is  manifest  that  if  the  least  or  third  part  were 
known,  the  remaining  parts  could  be  easily  ascertained  ; 
therefore, 

.  12* 


126  SOLUTION  OF  PROBLEMS 

Let  the  least  or  third  part     .     .     z=x. 
Then  the  ^econc?  part        .     .     .     =j;4-115. 
.'.  the  ^reaic^^  or ^r^i  part    .     .     =:a?4"115  +  180. 
But  the  sum  of  the  three  parts  .     =890. 

.-.  3ir+115  +  115-f  180=890, 

or  3a:+410  =  890; 

/.by  transposition,  3a;=890— 410, 

or  3a;=480, 

.'.  a;=:160=:  least  part. 

.•.a;4-115  =  160+ 115=275=  second  pan. 

and  a;4-115+ 180=160+115  +  180  =  455=:  greatest  pait. 

Prob.  22.  A  prize  of  2329  dollars  was  divided  between  two 
persons  A  and  B,  whose  shares  therein  were  in  proportion  of 
5  to  12.     What  was  the  share  of  each  ? 

Let  5jc=:A's  share  ; 
then  12a;=B's  share  ; 

.-.  5a.'+12a:=2329,  or  17a:=2329  ; 

and  a;=137 ; 

.-.their  shares  were  685  and  1644  dollars  respectively. 

Prob.  23.  A  fish  was  caught,  whose  tail  weighed  91bs. ; 
his  head  weighed  as  much  as  his  tail,  and  half  his  body ;  and 
his  body  weighed  as  much  as  his  head  and  tail.  What  did  the 
fish  weigh  1 

Let  2x=  the  number  of  lbs.  the  body  weighed  ;  then  9-\-x 
=  the  weight  of  the  tail ; 

.-.  9  +  9  +  ir=2ir  ; 

by  transposition,  a;  =  1 8  ; 

.-.  the  fish  weighed  36+27  +  9=72lbs. 

Prob.  24.  A  hare,  50  of  her  leaps  before  a  greyhound, 
takes  4  leaps  to  the  greyhound's  three  ;  but  two  of  the  grey- 
hound's leaps  are  as  much  as  three  of  the  hare's.  How  many 
leaps  must  the  greyhound  take  to  catch  the  hare  ? 

Let  3x=  the  number  of  leaps  the  greyhound  must  take  ; 
/.  4x=  the  number  the  hare  takes  in  the  same  time, 
.-.  4a;+50=  the  whole  number  she  takes, 

and  2  :  3  :  :  3j;  :  4a;+50  ; 
.♦.  9a;=8x+100  ; 
by  transposition,  a;=100, 
and  the  greyhound  must  take  300  leaps. 

Prob.  25.  The  number  of  soldiers  of  an  army  is  such,  that 
its  triple  diminished  by  1000,  is  equal  to  its  quadruple  aug- 
mented by  2000.     What  is  this  number  ? 


PRODUCING  SIMPLE  EQUATIONS.         127 

Let  X  designate  the  number  required  ; 
then,  we  are  conducted  to  this  equation, 

3^5—1000=4^:4-2000,  whence  «:=— 3000, 
which  gives  an  absurd  answer  with  respect  to  the  terras  of  the 
question,  since  that  a  number  of  soldiers  cannot  be  negative. 

181.  We  shall  render  this  impossibility  very  plain,  by  ob- 
serving that  the  triple  of  a  number  being  less  than  the  quadru- 
ple of  thp  same  number,  the  triple  diminished  by  1000' is  much 
less  than  the  quadruple  augmented  by  2000.  But  by  writing 
— X  in  the  place  of  -|-^  in  the  equation  of  the  problem,  then 
changing  the  signs  of  both  sides,  we  find 

3a;+1000  =  4a;— 2000;   .-.  a;:=3000. 
We  can  from  the  equation 

3a;+1000=:4a;— 2000, 

re-establish  the  enunciation  of  the  problem  in  such  a  manner 

that  there  results  from  the  solution  an  absolute  number,  that  is, 

a;=3000. 

If  in  place  of  taking  x  for  the  representation  of  the  unknown 

number,  we  had  taken 

3a;  — 6000,  or  a:=a;'— 6000 
we  should  find  for  the  equation 

a:'  — 19000==4a;'— 22000  ; 
.-.by  transposition,  22000  — 19000  =  4a;'  — 3a;', 
and  .-.  a;' =  3000  as  before. 


A  I  M  I  A' 

Thus  the  value  a;  =  —  3000  being  represented,  on  a  line,  by 
the  length  A^M,  counted  from  A'  towards  M,  or  to  the  left  of 
A^,  we  pass  by  the  substitution  a;=a;^— 6000  from  the  origin 
A^  to  the  origin  A,  to  the  left  of  A',  and  distant  from  A'  by 
6000=2A'M  ;  then  the  length  AM=a/  is  positive. 

Prob.  26.  A  Courier  sets  out  from  Trenton  for  Washington, 
and  travels  at  the  rate  of  8  miles  an  hour  ;  two  hours  after  his 
departure  another  Courier  sets  out  after  him  from  New-York, 
supposed  to  be  68  miles  distant  from  Trenton,  and  travels  at 
the  rate  of  12  miles  an  hour.  How  far  must  the  second  Cou- 
rier travel  before  he  overtakes  the  first  1 

N ■■ W 

T        R  M 

Let  X  represent  the  number  of  miles  which  the  second  Cou- 
irier  travels  before  he  overtakes  the  first ;  then,  by  a  little  at- 
tention, we  discover  that  this  distance  should  be  equal  to  the 
distance  from  New-York  to  Trenton,  or  NT =68  miles,  plus 


128  SOLUTION  OF  PROBLEMS 

the  distance  travelled  by  the  first  Courier  in  two  hours  which 

his  departure  preceded  that  of  the  second,  together  with  the 

number  of  miles  which  the  first  travels  whilst  tjie  second 

Courier  is  on  route  ;  that  is,  NM,  or  a;=NT  +  TR+RM. 

Let  us  translate  the  two  last  distances,  that  is,  TR  and  RM  ; 

in  the  first  place,  2  x  8zi=  16=TR=  the  number  of  miles  which 

the  first  Courier  travels  before  the  second  sets  out ;  then,  in 

order  to  find  an  expression  for  MR  ;  we  shall  say,  gince  the 

distances  passed  over  in  an  hour  are  as  8  :  12,  or  2  :  3  ;  as,  2  : 

2x 
3  : :  MR:  a;;  and  consequently  MR =—.     So  that  we  obtain 

for  a  translation  of  the  enunciation, 

^=,68+16+1=84+1^; 

by  multiplication,  docz='2^2-\-2x  ;  .*.  a:=i:252, 

that  is  to  say,  the  two  Couriers  would  meet  when  the  second 

shall  have  travelled  252  miles.     In  fact,  while  the  second 

2x 
travelled  252  miles,  the  first  travelled  168  miles  ;  since  —  is 

o 

the  expression  for  the  number  of  miles  which  the  first  travelled 

while  the  second  was  on  route  ;  that  is,  substituting  252  for  a;, 

2x     2X252     504     ^^^     .. 
-=-3— ^-3-^168  miles. 

Now,  the  place  from  whence  the  first  Courier  departed,  be- 
ing 68  miles  distant  from  New- York,  besides  he  has  the  ad- 
vantage of  having  travelled  16  miles  before  the  other  set  out. 
Consequently  684-16-f-168  must  be  equal  to  the  number  of 
miles  which  the  second  Courier  travels  before  they  meet ; 
that  is,  68  +  16  +  168=252. 

We  see  here  an  example  of  verification  of  the  value  of  the 
unknown  ;  it  is  a  proof  which  the  student  can,  and  should  al- 
ways make. 

182.  In  order  to  have  a  general  solution  of  this  problem, 
let  us  therefore  represent  in  general,  by  a  the  distance  between 
the  two  places  of  departure,  which  was  68  miles  in  the  preced- 
ing question,  by  b  the  number  of  hours  which  the  departure  of 
the  first  precedes  that  of  the  second,  by  c  the  number  of  miles 
that  the  first  Courier  travels  per  hour,  and  by  S  the  number 
which  the  second  travels  in  the  same  time.  Let  x=  the  dis- 
tance which  the  second  Courier  must  travel  before  they  meet ; 
then,  we  shall  have  the  distance  travelled  by  the  first  Courier 
during  the  time  that  the  second  has  been  travelling,  by  calcu- 
lating the  fourth  term  of  a  proportion  that  commences  thus  ; 


PRODUCING  SIMPLE  EQUATIONS.  129 

cXx      ex 
a  :  c  :  :  X  :  — r—  or  — . 
a         a 

The  first  Courier  travelling  c  miles  an  hour,  he  will  have  tra- 
velled cxb  miles  before  the  second  set  out. 

Therefore  by  the  condition  of  the  problem,  we  shall  have 

ca?   ,  ,     ,  ,  d{cb+a) 

x=  -—-{'hc-{-a  ;  whence  a:=— -j 

a  a — c 

which  gives  the  solution  of  all  questions  of  the  same  kind. 

In  order  to  show  the  use  of  this  formula,  let  us  resume  again 
the  preceding  enunciation,  and  by  recollecting  that  we  must 
replace  a  by  68,  6  by  2,  c  by  8,  and  d  by  12. 

Then  the  value  of  x  becomes 

x=  — ^=252  miles  as  before. 

12--6 

Prob.  27.  What  two  numbers  are  those,  whose  difference 
is  10,  and  if  15  be  added  to  their  sum,  the  whole  will  be  43  ? 

Ans  9  and  19. 

Prob.  28,  What  two  numbers  are  those,  whose  difference 
is  14,  and  if  9  limes  the  lesser  be  subtracted  from  six  times  the 
greater,  the  remainder  will  be  33  ?  Ans,  17  and  31. 

Prob.  29.  What  number  is  that,  which  being  divided  by  6, 
and  2  subtracted  from  the  quotient,  the  remainder  will  be  2  ? 

Ans.  24. 

Prob.  30  What  two  numbers  are  those,  whose  difference 
is  14,  and  the  quotient  of  the  greater  divided  by  the  lesser  3  ? 

Ans.  21  and  7. 

Prob.  31.  What  two  numbers  are  those,  whose  sum  is  60, 
and  the  greater  is  to  the  lesser  as  9  to  3  ?         Ans,  45  and  15. 

Prob,  32.  What  number  is  that,  which  being  added  to  5, 
and  also  multiplied  by  5,  the  product  shall  be  4  times  the  sum  ? 

Ans.  20. 

Prob,  33.  What  number  is  that,  which  being  multiplied  by 
12,  and  48  added  to  the  product,  the  sum  shall  be  18  times  the 
number  required  ?  Ans.  8. 

Prob.  34  What  number  is  that,  whose  -J  part  exceeds  its  J 
part  by  32  ?  Ans,  640. 

Prob.  35.  A  Captain  sends  out  ^  of  his  men,  plus  10  ;  and 
there  remained  J,  minus  15  ;  how  many  had  he  ?     Ans.  150. 

Prob,  36.  What  number  is  that,  from  which  if  8  be  sub- 
tracted, three-fourths  of  the  remainder  will  be  60  ?     Ans.  88. 

Prob.  37,  What  number  is  that,  the  treble  of  which  is  as 
much  above  40,  as  its  half  is  below  51  ?  Ans  26. 

Prob,  38.  What  number  is  that,  the  double  of  which  ex- 
ceeds four-fifths  of  its  half  by  40  ?  Ans  25. 


130  SOLUTION  OF  PROBLExMS    • 

Prob.  39.  At  a  certain  election,'  946  men  voted,  and  tho 
candidate  chosen  had  a  majority  of  558.  How  many  men  voted 
for  each  ?  Ans.  194  for  one,  and  752  for  the  other. 

Prob.  40.  After  paying  away  i  of  my  money,  and  then  i  of 
the  remainder,  I  had  140  dollars  left :  what  had  I  at  first  ? 

Ans.  180  dollars; 

Prob.  41.  One  being  asked  how  old  he  was,  answered,  that 
the  product  of  Jj  of  the  years  he  had  lived,  being  multiplied 
by  f  of  the  same,  would  be  his  age.     What  was  his  age  ? 

Ans.  30. 

Prob.  42.  After  A  had  lent  10  dollars  to  B,  he  wanted  8 
dollars  in  order  to  have  as  much  money  as  B  ;  and  together 
they  had  60  dollars.     What  money  had  each  at  first  ? 

Ans.  A  36  and  B  24. 

Prob.  43.  Upon  measuring  the  corn  produced  by  a  field, 
being  48  bushels  ;  it  appeared  that  it  yielded  only  one  third 
part  more  than  was  sown.     How  much  was  that  ? 

Ans.  36  bushels. 

Prob.  44.  A  Farmer  sold  96  loads  of  hay  to  two  persons. 
To  the  first  one  half,  and  to  the  second  one  fourth  of  what  his 
stack  contained.     How  many  loads  did  that  stack  contain  ? 

Ans.  128  loads. 

Prob.  45-  A  Draper  bought  three  pieces  of  cloth,  which 
together  measured  159  yards.  The  second  piece  was  15  yards 
longer  than  the  first,  and  the  third  24  yards  longer  than  the 
second.     What  was  the  length  of  each  ? 

Ans.  35,  50  and  74  yards  respectively. 

Prob.  46.  A  cask  which  held  146  gallons,  was  filled  with 
a  mixture  of  brandy,  wine,  and  water.  In  it  there  were  15 
gallons  of  wine  more  than  there  were  of  brandy,  and  as  much 
water  as  both  wine  and  brandy.  What  quantity  was  there  of 
each  ?  Ans.  29,  44,  and  73  gallons  respectively. 

Prob.  47.  A  person  employed  4  workmen,  to  the  first  of 
■whom  he  gave  2  shillings  more  than  to  the  second  ;  to  the  se- 
cond 3  shillings  more  than  to  the  third  ;  and  to  the  third  4 
shillings  more  than  to  the  fourth.  Their  wages  amounted  to 
32  shillings.     What  did  each  receive  ? 

Ans.  12,  10,  7,  and  3  shillings  respectively. 

Prob.  48.  A  Father  taking  his  four  sons  to  school,  divided 
a  certain  sum  among  them.  Now  the  third  had  9  shillings 
more  than  the  youngest ;  the  second  12  shillings  more  than 
the  third  ;  and  the  eldest  18  shillings  more  than  the  second; 
and  the  whole  sum  was  6  shillings  more  than  7  times  the  sum 
which  the  youngest  received.     How  much  had  each  ? 

Ans.  21,  30,  42,  and  60  shillings  respectively 


PRODUCING  SIMPLE  EQUATIONS.         131 

Prob.  49.  It  is  required  to  divide  the  number  99  into  five 
such  parts,  that  the  first  may  exceed  the  second  by  3  ;  be  less 
than  the  third  by  10  ;  greater  than  the  fourth  by  9  ;  and  less 
than  the  fifth  by  16.  Ans.  17,  14,  27,  8,  and  33. 

Prob.  50.  Two  persons  began  to  play  with  equal  sums  of 
money  ;  the  first  lost  14  shillings,  the  other  won  24  shillings, 
and  then  the  second  had  twice  as  many  shillings  as  the  first. 
What  sum  had  each  at  first  ?  Ans.  52  shillings. 

Prob.  51.  A  Mercer  having  cut  19  yards  from  each  of 
three  equal  pieces  of  silk,  and  17  from  another  of  the  same 
length,  found  that  the  remnants  together  were  142  yards. 
What  was  the  length  of  each  piece  ?  Ans.  54  yards. 

Prob.  52.  A  Farmer  had  two  flocks  of  sheep,  each  con- 
taining the  same  number.  From  one  of  these  he  sells  39,  and 
from  the  other  93  ;  and  finds  just  twice  as  many  remaining  in 
one  as  in  the  other.  How  many  did  each  flock  originally 
contain?       ,  Ans.  147 

Prob.  53.  A  Courier,  who  travels  60  miles  a  day,  has  been 
dispatched  five  days,  when  a  second  is  sent  to  overtake  him, 
in  order  to  do  which  he  must  travel  75  miles  a  day.  In  what 
\irae  will  he  overtake  the  former  ?  Ans.  20  days. 

Prob.  54.  A  and  B  trade  with  equal  stocks.  In  the  first 
year.  A  tripled  his  stock,  and  had  $27  to  spare  ;  B  doubled 
his  stock,  and  had  $153  to  spare.  Now  the  amount  of  both 
their  gains  was  five  times  the  stock  of  either.  What  was 
that  ?  Ans.  90  dollars. 

Prob.  55.  A  and  B  began  to  trade  with  equal  sums  of  mo- 
ney. In  the  first  year  A  gained  40  dollars,  and  B  lost  40  ; 
but  in  the  second  A  lost  one-third  of  what  he  then  had,  and  B 
gained  a  sum  less  by  40  dollars,  than  twice  the  sum  that  A 
had  lost ;  when  it  appeared  that  B  had  twice  as  much  money 
as  A.     What  money  did  each  begin  with  ?     Ans.  320  dollars. 

Prob.  56.  A  and  B  being  at  play,  severally  cut  packs  of 

cards,  so  as  to  take  off  more  than  they  left.  Now  it  happened 

that  A  cut  oft'  twice  as  many  as  B  left,  and  B  cut  off  seven 

times  as  many  as  A  left.     How  were  the  cards  cut  by  each  ? 

Ans.  A  cut  off  48,  and  B  cut  oft"  28  cards. 

Prob.  57.  What  two  numbers  are  as  2  to  3  ;  to  each  of 
which  if  4  be  added,  the  sums  will  be  as  5  to  7  ? 

Ans.  16  and  24. 

Prob.  58.  A  sum  of  money  was  divided  between  two  per- 
sons, A  and  B,  so  that  the  share  of  A-  was  to  that  of  B  as  5  to 
3  ;  and  exceeded  five-ninths  of  the  whole  sum  by  50  dollars. 
What  was  the  share  of  each  person  ? 

Ans.  450.  and  270  dollars. 


13a  SOLUTION  OF  PROBLEMS 

Prob.  59.  The  joint  stock  of  two  partners,  whose  particu- 
lar shares  differed  by  40  dollars,  was  to  the  share  of  the  les- 
ser as  14  to  5.     Required  the  shares. 

Ans.  the  shares  are  90  and  50  dollars  respectively. 

Prob.  60.  A  Bankrupt  owed  to  two  creditors  1400  dollars  ; 
the  difference  of  the  debts  was  to  the  greater  as  4  to  9.  What 
were  the  debts  ?  Ans.  900,  and  500  dollars. 

Prob.  61.  Four  places  are  situated  in  the  order  of  the  four 
letters. A,  B,  C,  D.  The  distance  from  A  to  D  is  34  miles, 
the  distance  from  A  to  B  :  distance  from  C  to  D  :  :  2  :  3,  and 
one-fourth  of  the  distance  from  A  to  B  added  to  half  the  dis- 
tance from  C  to  D,  is  three  times  the  distance  from  B  to  C. 
What  are  the  respective  distances  ? 

Ans.  AB  =  12,  BC  =  4,  and  00  =  18  miles. 

Prob.  62.  A  General  having  lost  a  battle,  found  that  he  had 
only  half  his  army  plus  3600  men  left,  fit  for  action  ;  one-eighth 
of  his  men  plus  600  being  wounded,  and  the  rest,  which  were 
one-fifth  of  the  whole  army,  either  slain,  taken  prisoners,  or 
missing.     Of  how  many  men  did  his  army  consist  ? 

Ans.  24000. 

Prob.  63.  It  is  required  to  divide  the  number  91  into  two 
such  parts  that  the  greater  being  divided  by  their  difference, 
the  quotient  may  be  7.  Ans.  49  and, 42. 

Prob.  64.  A  person  being  asked  the  hour,  answered  that  it 
was  between  five  and  six ;  and  the  hour  and  minute  hands 
were  together.     What  was  the  time  ? 

Ans.  5  hours  27  minutes  16^  seconds. 

Prob.  65.  Divide  the  number  49  into  two  such  parts,  that 
the  greater  increased  by  6  may  be  to  the  less  diminished  by 
11  as  9  to  2.  Ans.  30  and  19. 

Prob.  66.  It  is  required  to  divide  the  number  34  into  two 
such  parts  that  the  difference  between  the  greater  and  18, 
shall  be  to  the  difference  between  18  and  the  less  :  :  2  :  3. 

Ans.  22  and  12. 

Prob.  67.  What  number  is  that  to  which  if  1,  5,  and  13,  be 
severally  added,  the  first  sum  shall  be  to  the  second,  as  the 
second  is  to  the  third.  Ans.  3. 

Prob.  68.  It  is  required  to  divide  the  number  36  into  three 
such  parts,  that  one-half  of  the  first,  one-third  of  the  second, 
and  one-fourth  of  tlie  third,  shall  be  equal  to  each  other. 

Ans.  8,  12,  and  16. 

Prob.  69.  Divide  the.  number  116  into  four  such  parts, 
that  if  the  first  be  increased  by  5,  the  second  diminished  by 
4,  the  third  multiplied  by  3,  and  the  fourth  divided  by  2,  the 
result  in  each  case  shall  be  the  same.     Ans.  22,  31,  9,  and  54. 


PRODUCING  SIMPLE  EQUATIONS.       •iSS 

Prob.  70.  A  Shepherd,  in  lime  of  war,  was  phmdered  by  a 
party  of  soldiers  who  took  i  of  his  flock,  and  ^  of  a  sheep  ; 
another  party  took  from  him  ^  of  what  he  had  left,  and  J  of  a 
sheep  ;  then  a  third  party  took  ^  of  what  now  remained,  and 
J  of  a  sheep.  After  which  he  had  but  25  sheep  left.  How 
many  had  he  at  first  ?  Ans.  103. 

Prob.  71.  A  Trader  maintained  himself  for  3  years  at  the 
expense  of  50/.  a  year  ;  and  in  each  of  those  years  augmented 
that  part  of  his  stock  which  was  not  s'o  expended  by  J  thereof. 
At  the  end  of  the  third  year  his  original  stock  was  doubled. 
What  was  that  stock  ?  Ans.  740/. 

Prob.  72.  In  a  naval  engagement,  the  number  of  ships  ta- 
ken was  7  more,  and  the  number  burnt  2  fewer,  than  the  num- 
ber sunk.  Fifteen  escaped,  and  the  fleet  consisted  of  8  times 
the  number  sunk.     Of  how  many  did  the  fleet  consist  ? 

Ans.  32. 

Prob.  73,  A  cistern  is  filled  in  twenty  minutes  by  three 
pipes,  one  of  which  conveys  10  gallons  more,  and  the  other  5 
gallons  less,  than  the  third,  per  minute.  The  cistern  holds  820 
gallons.     How  much  flows  through  each  pipe  in  a  minute  1 

Ans.  22,  7,  and  12  gallons. 

Prob.  74,  A  sets  out  from  a  certain  place,  and  travels  at 
the  rate  of  7  miles  in  five  hours  ;  and  eight  hours  afterwards 
B  sets  out  from  the  same  place,  and  travels  the  same  road  at 
the  rate  of  5  miles  in  three  hours.  How  long,  and  how  far, 
must  A  travel  before  he  is  overtaken  by  B  ? 

Ans.  50  hours,  and  70  miles. 

Prob.  75.  There  are  two  places,  154  miles  distant,  from 
which  two  persons  set  out  at  the  same  time  to  meet,  one  tra.- 
velling  at  the  rate  of  3  miles  in  two  hours,  and  the  other  at 
the  rate  of  5  miles  in  four  hours.  How  long,  and  how  far,  did 
each  travel  before  they  met  ? 

Ans.  56  hours  i  and  84,  and  70  miles. 


13 


134 


CHAPTER  V. 

ON 

SIMPLE  EQUATIONS, 

INVOLVING  TWO  OR  MORE  UNKNOWN  aUANTITIES. 

183.  It  has  been  observed  (Art.  159),  that  an  equation  was 
the  translation  into  algebraic  language  of  two  equivalent 
phrases  comprised  in  the  enunciation  of  a  question  ;  but  this 
question  may  comprehend  in  it  a  greater  number,  and  if  they 
are  well  distinguished  two  by  two,  and  independent  of  one  an- 
other, they  furnish  a  certain  number  of  equations. 

Thus,  for  example,  let  us  propose  to  find  two  numbers,  such 
that  double  the  first  added  to  the  second,  gives  24,  and  that  five 
times  the  first,  plus  three  times  the  second,  make  65.  We  find 
here  two  phrases,  which  express  the  same  thing  in  different 
terms  ;  1st,  the  double  of  an  unknown  number,  plus  another  un- 
knoimt  number,  then  the  equivalent  24  ;  2d,  five  times  the  first  un- 
known number,  plus  three  times  the  second,  then  the  equivalent  65. 

The  translation  is  easy,  and  it  gives  these  two  determinate 
equations  : 

2a:+y:=i24  ;  5ar+ 3y=65. 

When  two  or  more  equations,  involving  as  many  unknown 
quantities,  are  independent  of  one  another,  they  are  called  c?e- 
terminate.  But  if  for  the  second  of  these  two  conditions  we 
had  substituted  this  :  and  such  that  six  times  the  first  number, 
plus  three  times  the  second,  make  72  ;  these  two  phrases  ex- 
press nothing  more  than  the  first  two,  since  that  we  have  only 
tripled  two  equal  results  ;  we  should  have  but  one  translation, 
and  consequently  a  single  equation.  It  can  therefore  happen 
tha^  we  may  have  less  equations  than  unknown  quantities, 
and  then  the  question  is  said  to  be  indeterminate  ;  because  the 
number  of  conditions  would  be  insufiicient  for  the  determina- 
tion of  the  unknown  quantities,  as  we  shall  see  clearly  illus- 
trated in  the  following  section. 

§  I.     ELIMINATION  OF  UNKNOWN  QUANTITIES    FR03V  ANY  NUM- 
BER OF  SIMPLE  EQUATIONS. 

184.  Elimination  is  the  method  of  exterminating  all  the  un- 
known quantities,  except  one,  from  two,  three,  or  more  given 


SIMPLE  EQUATIONS.  135 

equations,  in  order  to  reduce  them  to  a  single,  or  final  equa- 
tion, which  shall  contain  only  the  remaining  unknown,  and 
certain  known  quantities. 

185.  In  order  to  simplify  the  calculations,  by  avoiding  frac- 
tions, we  shall  here  make  use  of  literal  equations,  which  will 
modify  the  process  of  elimination  :  And  also,  to  avoid  the  in- 
convenience arising  from  the  multitude  of  letters  which  must 
be  employed  in  order  to  represent  the  given  quantities,  when 
the  number  of  equations  involving  as  many  unknown  quanti- 
ties surpasses  two,  we  shall  represent  by  the  same  letter  all 
the  coefficients  of  the  same  unknown  quantity ;  but  we  shall 
affect  them  with  one  or  more  accents,  in  order  to  distinguish 
them,  according  to  the  number  of  equations. 

186.  In  the  first  place,  any  two  simple  equations,  each  in- 
volving the  same  two  unknown  quantities,  may,  in  general,  be 
written  thus : 

ax-\-by=:c (A), 

a'x-\-b'y=zc'   ......     (B). 

The  coefficients  of  the  unknown  quantity  x  are  represent- 
ed both  by  a  ;  those  of  y  by  i  ;  but  the  accent,  by  which  the 
letters  of  the  second  equation  are  affected,  shows  that  we  do 
not  regard  them  as  having  the  same  value  as  their  correspond- 
ents in  the  first.  Thus  a!  is  a  quantity  different  frOm  a,  b'  a 
quantity  different  from  b, 

187.  We  can  readily  see,  by  a  few  examples,  how  any  two 
simple  ecjuations,  each  involving  the  same  two  unknown  quan- 
tities, may  be  reduced  to  the  above  form. 

Ex.   1.  Let  the  two  simple  equations, 
5x-\-2y^b—y—2x+7, 
9a:— 2y4-3=:a;— 7y4-16, 
be  reduced  to  the  form  of  equations  (A)  and  (B). 
By  transposition,  these  equations  become 
5a;+3y— y-f2a;=7-f-5, 
Qx—2y—x  +  ly  —  \6  —  ^; 
by  reduction,  we  shall  have 

7a;4-2y=12, 
8a;  +  5y=13; 
equations  which  are  reduced  to  the  form  of  (A)  and  (B),  and 
which  may  be  expressed  under  the  form  of  the  same  literal 
equations,  by  substituting  a,  6,  and  c,  for   7,  2,  and  12  ;  and 
a\  b',  and  c^,  for  8,  5,  and  13. 

Ex.  2.  Let  the  two  simple  equations, 

ma?  4- 6y — 7  3= pa: — 2y -f  3 , 
ra;— 9y+6=:3y  — 3a;+12, 
be  reduced  to  the  form  of  equations  (A)  and  (B). 


136  SIMPLE  EQUATIONS. 

By  transposition,  these  equations  become 
mx-{-6y — px-\-2i/ =  3-^7, 
rx—9y—3i/-{-3x=l2—6 ; 
by  reduction,  we  shall  have 

(m—p)x-{-   8y=10, 
(r+3)a:-12y=6; 
which  are  reduced  to  the  form  required,  and  which  may  be 
expressed  under  the  form   of  the  same  literal  equations,  by 
substituting  a  for  m—p,  b  for  8,  c   for  10,  a^  for  r-\-3,  b'  for 
—  12,  and  c'  for  6. 

In  like  manner  any  two  simple  equations  may  be  reduced  to 
the  form  of  equations  (A)  and  (B)  ;  hence  we  may  conclude 
that  a,  b,  c,  a,  b',  and  c',  may  be  any  given  numbers  or  quan- 
tities whatever,  positive  or  negative,  integral  or  fractional. 

It  is  to  be  always  understood,  that  when  we  make  use  of 
the  same  letters,  marked  with  different  accents,  they  express 
different  quantities.  Thus,  in  the  following  equations,  a,  </, 
a",  are  three  different  quantities  ;  and  the  same  of  others. 

188.  Any  three  simple  equations,  each  involving  the  same 
three  unknown  quantities,  may  be  expressed  thus  ; 

ax-{-bi/-{-czz=d     ....     (C), 
a'x-^b'y-\-c's=^d'       .     .     .     (D), 
a".x+b"y^(/'z=d^'^      ..(E); 
where  a,  b,  c,  d,  a\  b\  c'  d\  a'\  b" ,  c",  d" ,  are  known  quanti- 
ties^ and  a?,  y,  z,  unknown  quantities  whose  values  may  be 
found  in  terms  of  the  known  quantities. 

In  like  manner,  any  four  simple  equations  may  be  expressed 
thus ; 

ax-^by-\-cz-{-du=:e  .  .  .  .  (F), 
a'x-\-b'y-}-c'z-\-d'uz=ze'  .  .  .  (G), 
a'^x+b''y-\-c"x-\-d''u=e''  .  .  (H), 
a'''x+  b'"y-itc"'z-^d"'u  =  e'"  .  .  (I)  ; 
And  so*  on  for  five,  or  more  simple  equations. 

189.  Analysts  make  use  of  various  methods  of  eliminating 
unknown  quantities  from  any  number  of  equations,  so  as  to 
have  a  final  equation  containing  only  one  of  the  unknown 
quantities  ;  some  of  which  are  only  applicable  in  partic  ular 
cases ;  but  the  most  general  methods  of  exterminating  un- 
known quantities  in  simple  equations,  are  the  following. 

• 

FIRST  METHOD. 

190.  Let  US  consider,  in  the  first  place,  the  equations, 

ax-\-by=c     .     .     .     (A), 
afx-\-b'y=c\     .     .     (B). 


SIMPLE   EQUATIONS.  •      137 

It  is  evident  that  if  one  of  the  unknown  quantities,  x,  for 
example,  had  the  same  coefficient  in  the  two  equations,  it 
would  be  sufficient  to  subtract  one  from  the  other,  in  order 
to  exterminate  this  unknown  :  Let,  for  example,  the  equa- 
tions be 

\0x-{-Uy=27, 
10a;+   9y=15; 
if  the  second  be  subtracted  from  the  first,  we  shall  have 

lly-9y=:27~15,  or  2y=12. 
It  is  very  plain,  that  we  can  immediately  render  the  coeffi- 
cients of  X  equal,  in  the  equations  (A)  and  (B) ; 

By  multiplying  the  two  members  of  the  first  by  a^,  the  co- 
efficient of  a?  in  the  second  ;  and  the  two. members  of  the  se- 
cond by  a,  the  coefficient  of  x  in  the  first ;  we  shall  thus  ob- 
tain, ^ 
aax-\-a'hy=ia^c  ; 
adx'\-ab'y=.ac' . 
Subtracting  the  first  of  these  from  the  second,  the  unknown 
X  will  disappear,  we  shall  have  only 

{ah' —dli)y=.ac' — a''c, 
an  equation  which  contains  no  more  than  the  unknown  quan- 
tity y,  and  we  will  deduce  from  it 

ac'  —  a'c  *  ,  . 

"^W^-cTb    ■    ■    ■   (")• 

By  eliminating  in  the  same  manner  the  unknown  quantity 
y,  from  the  proposed  equations  ;  we  would  arrive  at  the  equa- 
tion 

(ab" — a'b)x = b'c — be' ; 
from  which  we  will  deduce 

b'c-bc'  ... 

"=^y3^     •     •     •     (^)- 

191.  The  process  which  we  have  just  employed,  may  be 
applied  to  all  simple  equations,  for  exterminating  any  number 
whatever  of  unknown  quantities. 

If  we  apply  this  process  to  three  equations,  involving  a?,  y, 
and  z,  we  will  at  first  eliminate  x  between  the  first  and  se- 
cond ;  then  between  the  second  and  third  ;  and  we  shall 
thus  arrive  at  two  equations,  which  involve  only  y  and  z,  and 
between  which  we  will  afterward  eliminate  y,  as  in  the  preced- 
ing article. 

If  we  effect  the  equation  in  z,  at  which  we  will  arrive,  we 
snail  have  a  factor  too  much  in  all  its  terms  ;  and  consequent- 
ly it  vjill  not  be  the  most  simple  which  might  be  obtained. 
13* 


138  SIMPLE  EQUATIONS. 

SECOND  METHOD. 

1 92.  Let  us  resume  again  the  equations, 
(A)     .     .      ca;+^y=c;  a'x-\-h'y=zc'     .     .     .  (B)  ; 
If  we  find  the  value  of  a;*  in 'terms  of  y  and  the  known  quan- 
tities in  each  of  these  equations,  we  shall  have 

c — by  c' — h'y 

X—  ^,  X— -^; 

a        •  a' 

the  equality  of  the  second  members,  furnishes  the  equation 
c — hy     c'  —  h'y 
a  a'      ' 

which,  by  making  proper  reductions,  gives 
ac' — a'c 
y^  ab^Yb ' 
by  substituting  this  value  for  y,  in  one  of  the  values  of  a:,  we 
shall,  after  the  reductions,  have 

b'c —  he' 
ab  — aib 
These  values  of  x  and  y  are  the  same  as  before. 
Now,  it  is  evident,  that  by  proceeding  in  the  ^ame  manner, 
with  three  equations  containing  x,  y,  and  z,  we  will  find  the 
value  of  X  in  each  of  them,  then  by  comparing  these  values, 
we  shall  arrive  at  two  equations,  involving  only  y  and  z,  from 
which  we  can  eliminate  y,  as  in  equations  (A)  and  (B).     And, 
we  can  proceed,  in  a  similar  manner,  when  there  are  four  equa- 
tions with  four  unknown  quantities  ;  and  so  on,  for  five,  or  more 
equations. 

THIRD  METHOD. 

1*93.  Now,  if  in  the  equation  (A),  we  find  the  value  of  x,  in 

terms  of  y  and  the  given  quantities,  we  shall  have 

c—by 

x=z -^  ; 

a 


by  substituting  this  value  in  equation  (B),  we  shall  have 

c— 

which,  by  reduction,  becomes 


,     c—by   ,  . 
a 


ac — a  c 


(ah'-a'b)y  =  ac'-a'c,  ..y=^^^__^,^  ; 

this  value  being  substituted  for  y  in  the  above  value  of  a?,  after 
making  the  proper  reductions,  we  shall  obtain 
_b'c—bc' 


(1) 

(2) 
(3) 


SIMPLE  EQUATIONS.  139 

These  values  of  x  and  y  are  the  same  as  in  the  two  former 
instances. 

194.  We  might  eliminate  in  like  mannef,  when  any  num- 
ber of  simple  equations  are  concerned  ;  thus,  for  example  : 
Let  it  be  required  to  deduce  from  the  three  equations,  (C),  (D), 
and  (E),  (Art.  188),  a  single  equation  involving  only  the  un- 
known quantity  z. 

By  finding  the  value  of  x  in  each  of  these  equations,  in 
terms  of  y,  z,  and  the  given  quantities,  we  shall  have 
_d'—by--cz 

~         a 

d'—h'y—dz 

x= ^, .     . 

a 

_d"-h''y-c'z 

a 

Putting  the  first  value  of  x  equal  to  the  second,  and  also 

equal  to  the  third,  we  shall  have  these  two  equations, 

d—by—c:^    d' — h'y—c'z 
—  f  , 

a  a 

d—by~cz  _d"-^h"y'^&'z 

~a         ~  of'  ' 

From  which  we  deduce,  by  ^eduction  and  proceeding  as  in 

equations  (A)  and  (B), 

icdc — ac'\z-\-ad' — a!d  ,  . 

y= W^^f^-  •    •   •   (4); 

(a"c—ac:)z+ad" —a!'d. 

y= — ab'^-^^h — -   •  •  (^)- 

The  equality  of  the  second  members  furnishes  the  equation 
{a'c—ac')z-^ad'—a'd     {a'^c—ac^')z-\-ad" — a'^'d 
ab'  —  a'b  ~  ab'f^a"b         ~ 

which,  by  proper  reductions,  will  give  the  value  of  z  :  having 
obtained  the  value  of  z,  substitute  it  in  equation  (4)  or  (5), 
and  the  value  of  y  can  be  readily  found. 

Now,  the  values  of  y  and  z  being  known,  by  substituting 
them  in  the  equation  (1),  (2),  or  (3)  ;  we  shall  easily  obtain 
the  value  of  x. 

FOURTH  METHOD. 

195.  Let,  as  before,  the  two  equations  be 

(A)  .  .  .  ax-{-by=:c  ;  a'x-{-yy=c'  .  .  .  (B).  * 
Multiplying  equation  (A)  by  some  indeterminate  quantity 
m,  it  will  become 

amx-\-bmy=mc'f 


140  SIMPLE  EQUATIONS. 

and  subtracting  from  this  result  equation  (B),  we  shall  have 
(am—a')x-\-(hm—h')yz=zcm — c'     .     .     .     (6). 
And  since  the  ^lue  of  w,  in  this  equation,  is  indeterminate, 

we  can  take  hm-~y=zO,  or  m=Y-;  in  which  case  the  second 

term  will  disappear,  we  shall  have 

b'      , 

cm — c  .         b         _cb'—bc' 
«=— — 7= — rr~ 


0 


aV — a!b 


which  is  the  same  value  of  a?,  as  before. 

Also,  as  the  value  of  a:,  thus  found,  is  independent  of  that 

of  m,  we  can  now  take  am=ia'  or  m=z-— ;  according  to  which 

supposition  the  terra  involving  x  will  vanish,  and  the  result 
will  give 

caf — dc' 
y'^ba'-'ah'' 
By  changing  the  signs  of  the  numerator  and  denominator 
(Art.  128)  of  this  value,  its  denominator  will  be  the  same  as 
that  of  X,  since  we  shall  have, 

which  is  the  same  value  of  y  as  in  each  of  the  preceding 
methods. 

This  method,  given  by  Bezout,  is  very  simple  for  elimi- 
nating all  the  unknown  quantities,  except  one  ;  besides,  it  has 
the  advantage  of  greater  brevity  above  the  preceding  methods, 
as  we  can  deduce  the  values  of  each  of  the  unknown  quanti- 
ties from  the  same  equation. 

§  II.     RESOLUTION  OF  SIMPLE  EQUATIONS, 

Involving  two  unknown  Quantities. 

196.  When  there  are  two  independent  simple  equations,  in- 
volving two  unknown  quantities,  the  value  of  each  of  them 
may  be  found  by  any  of  the  following  practical  rules,  which 
are  easily  deduced  from  the  Articles  in  the  preceding  Section. 

RULE  I. 

197.  Multiply  the  first  equation  by  the  coefficient  of  one  of 
the  unknown  quantities  in  the  second  equation,  and  the  se- 


SIMPLE  EQUATIONS.  141 

cond  equation  by  the  coefficient  of  the  same  unknown  quan- 
tity in  the  first.  If  the  signs  of  the  term  involving  the  un- 
known quantity  be  alike  in  both,  subtract  one  equation  from 
the  other  ;  if  unlike,  add  them  together,  and  an  equation  arises 
in  which  only  one  unknown  quantity  is  found. 

Having  obtained  the  value  of  the  unknovyn  quantity  from 
this  equation,  the  other  may  be  determined  by  substituting  in 
either  equation  the  value  of  the  quantity  found,  and  thus  re- 
ducing the  equation  to  one  which  contains  only  the  other  un- 
known quantity. 

Or,  multiply  or  divide  the  given  equations  By  such  numbers, 
or  quantities,  as  will  make  the  term  that  contains  one  of  the  un- 
known quantities  the  same  in  each-equation,  and  then  proceed 
as  before. 

Ex.  1.  Given   ^^[^g^^jM  to  find  the  values  of  a:  and  y. 

Multiply  the  1st  equation  by  5,  then  10x4-15y=115  ; 
2nd    .    .     .    .    2,  .    .    10a;—  4y=  20; 

.•.  by  subtraction,  19y= 95, 
95 
by  division,  y=— ;  .*.  y=5. 

Now.  from  the  first  of  the  preceding  equations,  we  shall  have 

23  — 3y     ,  .  .23  —  15     8      ^ 

oc=—^=(since  y=5)  — ^-^—=4. 

The  values  of  a;  and  y  might  be  found  in  a  similar  manner, 
thus : 

Multiply  the  1st  equation  by  2,  then  4a;4-6y=46  • 
2nd 3,    .    15a;— 6y=i30; 

/.by  addition,  19a?=76, 

by  division,  a?=:— -=4 

Now,  from  the  first  of  the  preceding  equations,  we  shall 

23— 2a;     ,  .  .x23  — 8     15      , 

nave  i/= — =(smce  a;=4)  — - — =—=5. 

o  o  o 

Ex.  2.  Given  |  e^i  12^=48' |  ^^  ^""^  ^^®  ^^^"®^  °^  * 
and  V- 

Multiply  the  1st  equation  by  6,  then  24a;+54y=210  ; 
2nd    ....    4,     .     24a;+48y  =  192; 

.*.  by  subtraction,  6y=  18, 


142  SIMPLE   EQUATIONS. 

by  division,  y=:-— =3. 

Now,  from  the  first  of  the  preceding  equations,  we  shall 

,  35  — 9y     ,.  ^,35-9x3     35-27 

have  x=z — - — -=(since  y=3) = , 

4  4  4 

or  oc=-j  r.x=2 

The  values  of  x  and  y  may  be  found  thus  ; 
Multiply  the  1st  equation  by  3,  then  12a;-f-27y=105  ; 
2nd     .     .     .      2,     .       12a:+24y=   96; 

,     /.  by  subtraction,  3y=9, 

by  division,  y=-=3. 
o 

.    J  35-27     8     ^ 

And  .'.x=: — =o=2. 

4  2 

The  numbers  3  and  2,  by  which  we  multiplied  the  given 
equations,  are  found  thus  ; 

The  product  of  two  numbers  or  quantities,  divided  by  their 
greatest  common  measure,  will  give  their  least  common  mul- 
tiple. 

.*.  ——-=12  the  least  common  multiple, 

12 
Then — =3,  the  number  by  which  the  first  equation  is 
4 

12 
multiplied  ;  and  -^=2,  the  number  by  which  the  second  equa- 
tion is  multiplied. 

By  proceeding  in  a  similar  manner  with  other  equations, 
the  final  equation  will  be  always  reduced  to  its  lowest  terms. 

Ex.  3.  Given   \  o^tl^^fS'  I  to   find  the   values    of  x 
(  3a;+7y=67,  S 

andy. 

Multiply  the  2nd  equation  by  5,  then  15a:+35y=335 ; 

Ist      .      .       .      3,     .      15a;+12y=l74; 

/.  by  subtraction,  23y  =  161, 

andy=--=7; 

whence,  5«=58—4y=58— 28=30, 

and  .*.  aj=-r-=o. 
5 


SIMPLE  EQUATIONS.  143 

If  the  second  equation  had  been  multiplied  by  4,  and  sub- 
tracted from  the  first  when  multiplied  by  7,  an  equation  would 
have  arisen  involving  only  x,  the  value  of  which  might  be  de- 
termined, and  thence,  by  substitution,  the  value  of  y. 

Ex.  4.  Given  |  g^Ze^^^lo'  \  ^^  ^^^  ^®  ^^^^®^  ^^  * 
and  y. 

Multiply  the  first  equation  by  3, 

18a;— 6y=     42  ; 
but  5a;— 6y=  — 10  ; 

•  -'-by  subtraction,  I3a;=52,  and  a;=4, 

5a;-M0     20+10     30     ^ 
whence  y=_^_=_^=_=5. 

198.  These  values  being  substituted  in  the  place  of  x  and 
y  in  each  of  the  equations,  shall  render  both  members  iden- 
tically  equal,  or,  what  is  the  same  thing,  each  of  the  equations 
will  reduce  to  0=0. 

Thus,  by  substituting  4  for  a;,  and  5  for  y,  in  the  above 
equations,  they  become 

6X4-2X5=      14.)     ^^5      14=      14  ;  > 

5X4-6X5=i:-10;  J     °^    ^  -10  =  — 10.   J 

Therefore,  by  transposition, 

,     14  — 14=0,  or  0  =  0; 
and  —10+10=0,  or  0=0. 

Since  (Art.  56)  14  —  14=0,  and  10—10=0. 

If  these  conditions  do  not  take  place,  it  is  evident  that  there 
must  be  an  error  in  the  calculation :  therefore,  the  student, 
whenever  he  has  any  doubt  respecting  the  answer,  should  al- 
ways make  similar  substitutions. 

Ex.  5.  Given    \  ll^+3y  =  100,  }    ^^  ^^^  ^^^  ^^j^^^  ^^  ^ 
^    4a;— 7y=      4,  S 
and  y. 

Mult,  the  1st  equation  by  7,  then  77aj+21y=700, 
2d      ...      3,      .     12a,— 21y=   12; 

.-.  by  addition,  89a;=712, 

712     • 
by  division,  x=-^  ; 

and  .-.  a;=8  ; 
whence  3y=100-lla;=100-ll  X8=100-88=12 ; 

.•.y=-3-=4. 


144  SIMPLE  EQUATIONS. 


-+7y=99, 
Ex.  6.    Given  <J  J>  to  find  the  values  of  x  and  y 

Multiply  each  equation  by  7, 

.-.  a;+49y=693, 
andy4-49a:=357; 

.-.  by  addition  50a:+50y=1050, 

1  ,-     J-  .  •  .         1050     ^, 

and  by  division,  x-\-y=———=21  ; 
oU 

but  since  a;+49y=693, 


Bubtracting  the  uppet  equation  from  the  lower, 

we  have  48y=693— 21=672, 

whence  a;=z21— y=21  — 14=7. 

x-\-2  \ 

■n     F^    /-<•     _     7      3  ^~       '  f  to  find  the  values  of « 

Ex.  7.  Given    <     ,^  >  „„  j 

v4-5  C  and  y. 

4  y 

Clearing  the  first  equation  of  fractions,     , 

a;4-2+24y==:93  ; 

.-.  by  transposition,  a;+24y=91     .     .     .     (1) 

Clearing  the  second  equation  of  fractions, 

y+5  +  40a;=768; 

.-.  by  transposition,  40a;+y=763     .     .     .     (2). 

Multiplying  equation  (1)  by  40,  and  subtracting  equation 

(2)  from  it, 

40a;+960y=3640; 

40a;-        y=   763; 

.-.  959y=2877, 

and  by  division,  y=3  ; 
From  equation  (1),  a;  =  91— 24y, 
#      .*.  by  substitution,  a;zz:91 — 24x3, 

or  a;=91— 72,  .-.  a;=19. 

If  from  equation  (2),  multiplied  by  24,  equation  (1)  had 
been  subtracted,  an  equation  would  have  arisen  involving  only 
a?,  the  value  of  which  might  be  determined,  and  this  being  sub- 
stituted in  either  of  the  equations,  the  value  of  y  might  also 
be  found. 


SIMPLE  EQUATIONS.  145 

Ex.  8.  Given  \  ^+y=f  I  to  find  the  values  of  x  and  y. 
f  a? — y — 0,  >         ^ 

By  addition,  2x=a-\-b  ;  .-.  a;=— ^- — 

By  subtraction,  2y=za — 5,  .-.  yn: . 

2 

Ex.  9.  Given    H^+g^^^^' ^  to  find  the  values  of  a;  and  y 

Multiply  the  1st  equation  by  2,  then  a?+4y=24  ; 
9    2nd  ';    .    .     .    2,     .     ic— 4y=:   8  ; 


by  addition,  2a; =32, 

32 
.'.  by  division,  x—-—=\Q. 

4> 


By  subtraction,  8y=!l6  ; 
,  .*.  by  division,    y=  2. 

Or,  the  values  of  x  and  y  may  be  found  thus  : 

From  the  first  equation  subtract  the  second,  and  we  have 

4y=8,  .•.y=2. 
Add  the  first  equation  to  the  second, 

and  .*.  a;=16. 

Ex.  10.  Given  4rr+3y=31,  and  3a:+2y=22  ;  to  find  the 

values  of  x  and  y.  Ans.  a:=4,  y=:5. 

Ex.  11.  Given  5a;— 4y=19,  and  4a?-f2y=36,  to  find  the 

values  of  x  and  y.  Ans.  x=7,  y=:4. 

Ex.  12.  Given  ^_2y=2,  and  ^-^Ij^yJ^  ;  to  find 

the  values  of^  and  y.  Ans.  a;=ll,  y=l. 


Ex.  13.  Given  5^-^+14=:18, 


to  find  the  values  of  x 


Ans.  a;=5,  and  y=2. 

Ex.  14.  Given?^±5y +^=8,)^^  ^^^  ^^^  ^^1^^^  ^^  ^ 

,  7y— 3a?  ,  ,   (and  y. 

and-i-^ y=ll») 


14 


Ans.  a; =6,  and  yac8. 


146  SIMPLE  EQUATIONS. 

Ex.  15.  Given  3a:+-^=22 

4i 


and  lly — ^=20, 


2x  \  ^^  ^"^  ^^®  values  of  x  andy. 

Ans.  a:i=5,  and  y=2. 
Ex.  16.  Given  a?+l  :  y  : :  5  :  3,  \ 

,  2a?     5— y_41     2a;— 1    >  to  find  the  values 

^"  3     2  ~i2     r~'i 

of  X  and  y. 

Ans.  a?=:4,  and  y=3. 

^    \»    r^-       ^—2       10— a?    y— 10  \ 
Ex.  17.  Given — =^ ,  J 

2y+4     2a:+y     a,+ 13,  >  to  finS  the  values 

and-^ ^=-4-3 

of  X  and  y. 

Ans.  a;  =  7,  and  y  =  10. 
Ex.  18.  Given  a:+15y=:53,  >  ,    ^   j  ,1        ,  r  1 

and  y  +   3^=27,  ]  ^"^  ^^  ^^^  ^^^"^'  ^^  ^  ^"^  y- 
Ans.  a;=8,  and  y=3. 
Ex.19.  Given  4a;4-  9y=51,  )      ^    , ,,         ,  n         , 

and  8x- 1 3y  =   9,  J  ^^  ^"^  ^^^  ^^^"^"  «^  ^  ^^^  y- 
Ans.  a:=6,  and  y=3. 

Ex.  20.  Given  |+t=6,  ) 
o  4  r 
^     ,,  >  to  find  the  values  of  x  and  v- 

a„d|+|=5f,^ 

Ans.  a;=12,  and  y=16. 

RULE  II. 

199.  Find  the  value  of  one  of  the  iinknown  quantities  in 
terms  of  the  other  and  known  quantities,  in  the«more  simple 
of  the  two  equations  ;  and  substitute  this  value  instead  of  the 
quantity  itself  in  the  other  equation  ;  thus  an  equation  is  ob- 
tained, in  which  there  is  only  one  unknown  quantity  ;  the  va- 
lue of  which  may  be  found  as  in  the  last  Rule. 

Ex.  1.  Given   J  3^+ ^^^^M  to  find  the  values  of  a?  andy. 

From  the  first  equation,  a?r=17— 2y ; 

Substituting  therefore  this  value  of  x  in  the  second  equation, 
3.(17-2y)-y=32, 
or  51 — 6y — y=2  ; 
by  changing  the  signs,  and  transposing ; 


SIMPLE  EQUATIONS.  147 

7y=51— 2  =  49, 
.'.  by  division,  y—^  ; 

whence  a:=  17— 2yr=  17—14=3. 
Here  a  value  of  y  might  be  determined  from  either  equa- 
tion, and  substituted  in  the  other ;  from  which  woukl  arise  an 
equation  involving  only  a:,  the  value  of  which  might  be  found  ; 
and  therefore  the  value  of  y  also  might  be  obtained  by  sub- 
stitutio)!,  thus  ; 

From  the  second  equation,  3/  =  3a;— 2  ;  substituting  there- 
fore this  vahie  of  y  in  the  first  equation  ;  we  have, 
a:  +  2  .  (3x— 2)  =  17, 
or  a;-f-6a;  — 4  =  17 ; 
.-.by  transposition,  7a;  =17 +4=21 

,.      V   •  •  21 

by  division,  a:  =  -— ,  .•.a;=3  ; 

and  .-.3^  =  30;— 2  =  3x3— 2  =  9— 2=7 

Ex.  2.  Given   \  ^^t^^^^L  .       \  to  ^^^  ^he  values  of  x 
>  5.r-f  10  =  78 +y,  S 

and  y. 

From  the  first  equation,  y=60  —  Sa?  ; 

Let  the  value  of  y  be  substituted  in  the  second  equation, 
and  it  becomes, 

5a:+10=78-f-(60-3a:). 

Then,  by  transposition,  8a;=78  +  60  — 10  ; 

128 
and  by  division,  a:=-— -=  16. 

Whence,  y=60  —  3x=60  — 3x16  =  60-48 

Ex.  3.  Given    ^     ^        ""     """'  ^    ^^  ^"^  '^^  ^^^"^^  ^^  ^ 


.•.y=12. 


3 

Mult,  the  1st  equation  by  3,  then 

a:-hy=198-6y  .  .  .  (1), 
2nd  by  3,  then  a;— y= 186  — 6x  .  .  .  (2); 
From  equation  (1),  we  have  a:=198 — 7y ; 

(2), 7a;-y=186; 

By  substituting  the  above  value  of  a:,  in  the  last  equation,  it 
becomes 

7(198- 7y)-y=186, 
or,  1386— 49y—y=  186; 
by  transposition,  —50y=186  — 1386  =  — 1200, 
by  changing  the  signs,  50y  =  1200, 


148  SIMPLE   EQUATIONS. 

.....  1200     „^ 

. .  by  division,  y=— — -=24. 

Whence,  a;=:198-7y=198— 7x24=198-168, 

.-.  a;=30. 

Ex.  4.  Given   <      ,    ^~c^' >  to  find  the  values  of  a;  and  y. 

From  the  second  equation,  a; =60 — y  : 
By  substituting  this  value  of  a?  in  the  1st  equation,  we  have, 

60— y4-2y=80, 
by  transposition,  y=80— 60, 

.•.y=20. 
And  a;=60— y=  (by  substitution)  60—20, 

.-.  07  =  40. 

Ex.  5.  Given   J  q^_  ^'Z.  o' M^  find  the  values  of  a;  and  y. 

From  the  1st  equation,  a;=17— 2y. 
And  this  value  substituted  in  the  second, 
.     3(17-2y)-y=2, 
or  51— 6y  — y=2, 

by  transposition,  &;c.,  7y=49, 

.-.  by  division,  y=7, 
whence,  a;=17-2y=17-2  x7  =  17-14, 

Ex.  6.  Given   <  ^2_  2Z5'  (  ^^  find  the  values  of  a;  and  y. 

From  the  first  equation,  a: =5  -y, 

squaring  both  sides,  a?2— .(5_yj2 
And  by  substituting  this  value  for  x^  in  the  second  equa- 
tion, it  becomes, 

(5-y)2-y2  =  5, 

by  reduction,  25 — 10y=5, 
by  transposition,  10y=20, 

.-.  by  division,  y=2. 
Whence,  a;=5— y  =  5  — 2  =  3. 

Ex.  r  Given   <  >   ^nd  y. 

^|+8x=131,)  ^ 

Multiplying  the  first  equation  by  8, 
a?-|--64y=1552, 

.'.by  transposition,  a;=1552— 64y. 
And  substituting  this  value  for  a;,  in  the  second  equation,  it 
becomes, 


SIMPLE  EQUATIONS.  149 

|+8(1552-64y)=:131, 
o 

by  reduction,  y+99328— 40963/=1048, 

by  transposition,  4095y=z: 98280, 

^     ,.  .  .  98280 

by  division,  y=^^^; 

.•.y=24. 
Whence  a;=1552  —  64y=  1552  — 64x24, 
or  0:=  1552  — 1536; 

.•.a;=16. 
The  value  of  y  might  be  found  from  the  second  equation,  in 
terms  of  x  and  the  known  quantities  ;  vi^hich  value  of  y  substi- 
tuted for  it  in  the  first,  an  equation  would  arise  involving  only 
a?,  the  value  of  which  might  be  found  ;  and  therefore  the  value 
of  y  also  may  be  obtained  by  substitution. 

Ex.  8.  Given  ?^±^=27,  and  ^^=6,  to  find  the  va- 

lues  of  a;  and  y. 

Ans.  jc=9,  and  y=:6. 

Ex.  9.  Given  15y4-4^a;=300,  and  a?+15y=36,to  find  the 
values  of  x  and  y. 

Ans.  a:=6,  and  y=:2. 

Ex.  10.  Given  3a;+y=:60,  and  5a;+10=78H-y,  to  find  the 
values  of  a?  and  y-  Ans.  a:=::16,  and  y=12. 

Ex.  11.  Given  10a;— 3y=38,  and  3a?— y=ll,  to  find  the 
the  values  of  x  and  y.  Ans.  a;— 5,  and  y=4. 

Ex.  12.  Givena;+y=:198  — 6y,anda;— y=186  — 6a;,  tofind 
the  values  of  x  and  y.  Ans.  a;=30,  and  y=24. 

Ex.  13.  Given  ^+^=26,  and  |4-8a;=131,  to  find  the  va- 

lues  of  X  and  y.  Ans.  a; =16,  and  y=24. 

X       11  X       11 

Ex.  14.  Given --1-^:^7,  and --+^=8,  to  find  the  values  of 
2     3  o     2 

X  and  y.  Ans.  a;r=6,  and  y=:13. 

Ex.  15.  Given  4a;4-y=34,  and  4y+aj=16,  to  find  the  va- 
lues of  X  and  y.  Ans.  a;=:8,  and  y=2. 

Ex.  16.  Given  3a;+2y=54,  and  a:  :  y  : :  4  :  3,  to  find  the 
values  of  x  and  y.  Ans.  a;=12,  and  y=9 

Ex.  17.  Given  ^±?-f6y=21,  and  ^^+5a;=23,  to  find 

the  values  of  x  and  y.  Ans.  a;=4,  and  y=3. 


14* 


150  SIMPLE   EQUATIONS. 


RULE  in. 

200.  Find  the  value  of  the  same  unknown  quantity  in  terms  of 
the  other  and  known  quantities,  in  each  of  the  equations  ;  then, 
let  the  two  values,  thus  found,  be  put  equal  to  each  other  ;  an 
equation  arises  involving  only  one  unknown  quantity  ;  the  va- 
lue of  which  may  be  found,  and  therefore,  that  of  the  other  un- 
known quantity,  as  in  the  preceding  rules. 

This  rule  depends  upon  the  well-known  axiom,  (Art.  47)  ; 
and  the  two  preceding  methods  are  founded  on  principles  which 
are  equally  simple  and  obvious. 

Ex.  1.  Given    |  2^t^^^lOo' (  ^^  ^"^   ^^®  ""^^"^^    *^^  ' 

and  y. 

From  the  first  equation,  a:=100— 3y, 

^^         ^  ,         100— y 

and  from  the  second,  x= — - — -  ; 

•••^=100-3,. 

Multiplying  by  2, 100— y=200-6y, 

by  transposition,  6y— y=200  — 100, 

or,  5y=  100; 

.'.  by  division,  y=:20, 

whence,  a;=  100— 3y=100  — 3x20; 

.•.a;=40. 
Here,  two  values  of  y  might  have  been  found,  which  would 
have  given  an  equation  involving  only  x  ;  and  from  the  solu- 
tion of  this  new  equation,  a  value  of  ar,  and  therefore  of  y, 
might  be  found. 

Ex.  2.  Given  ^aj+Jy =7,  and  ia;4-iy= 8,  to  find  the  values 
of  X  and  y. 

Multiplying  both  equations  by  6,  and  we  shall  have 
3a:+2y=42,  and  2a;-|-3y=48, 

42— 2y 
From  the  first  of  these  equations,  x=  — - — -^ 

o 

and  from  the  second,  x=  — - — -; 

42— 2y_48— 3y^ 
•■       3      ~"      2       * 
Multiplying  each  member  by  6,  we  shall  have 
84— 4y=114— 9y; 

by  transposition,  9y— 4y=144— 84, 
or5y=60;  .•.y=12 


SIMPLE  EQUATIONS.  151 

And,  by  substituting  this  value  of  y,  in  one  of  the  values  of 
Xi  the  first,  for  instance,  we  shall  have 

42— 24_  18_^ 


3  3 

Ex.  3.  Given  8a;+18y=94,  and  8a;— 13y=l,  to  find  the 
values  of  x  and  y. 

From  the  first  equation,  x= — ^—^  ; 

4 

and  from  the  second,  x=z — - — ^  : 

8 

47— yy__H-i3y 

•■•^T~~"~8      ' 

And  multiplying  both  sides  of  this  equation  by  8, 
94  — 18y=14-13y; 
.-.by  transposition,  —18y—13y=— 944-1  ; 
Changing  the  signs,  or  what  amounts  to  the  same  thing, 
multiplying  both  sides  by  —  1 ,  and  we  shall  have 

18y+13y=94-l,or31y=93; 

,                 l  +  13y     1  +  39     40     ^ 
whence  x= -= — ■ — = — =5. 

8  8  8 

From  the  first  equation,  x=a-^y  ; 

and  from  the  second,  a;= — - — ^  ; 
b 

de — cy 

•••«-y=— j-^; 

and  multiplying  by  b,  we  shall  have 
ab — by=:de — cy  ; 

by  transposition,  cy — by=^der—ab  ; 
by  collecting  the  coefficients,  {c-'b)y=de—ab  ; 

,      J.   .  .               de — ab 
.*.  by  division,  y= — ; 

,  de — ab 

whence  x=a—y=a r-; 

^  c — b 

ca—ab — de-^-ab  _ca — de 
e—b  "^  c—h 


that  is,  X: 


Ex.  5.  Given  3/r-|-7y=79,  and  2y—\x=9,  to  find  the  va- 
lues of  a:  and  y.  Ans.  «=10,  and  y=7. 


162  SIMPLE  EQUATIONS. 

Ex.  6.  Given^i^-f  1=6,  and^^4-3=4,   to   find  the 

values  of  a?  and  y.  Ans.  a:=:ll,  and  y=4. 

2/p 3  57 

Ex.  7.  Given  — hy=7,  and  5a:— 13y=r— ,to  find  the 

values  of  a?  and  y.  Ans.  a?=:8,  and  y=^ 

Ex.  8.  Given  !^=?fijH:l_  and  8  -  ^  =  6,  to 
3  5  5 

find  the  values  of  x  and  y.  Ans.  a; =13,  and  y=3. 

Ex.  9.  Given  a:+y=10,  and  2a;— 3y=5,  to  find  the  values 
of  X  and  y.  Ans.  a; =7,  and  y=3. 

Ex.  10.  Given  3a?— 5y=13,  and  2a:4-7y=:81,  to  find  the 
values  of  a?  and  y.  Ans.  a:=  16,  and  y=7. 

Ex.  11.  Given -^^+8^=31,  and  ?^ -t  10a;=192,  to 
find  the  values  of  x  and  y.  '        Ans.  a;=19,  and  y=3 

Ex.  12.  Given  ?^3^+14=18,  and?^^=3,  to  find  the 

values  of  x  and  y.  Ans.  a:=5,  and  y=2. 

Ex.  13.  Given— ^-^=8--,   i  ,     ^    ,   ,         ,  , 

6  3    r    to  find  the  values  of  x 

7y— 3a;  (   and  y. 

and-^- — =ll+y,\  ^ 

Ans.  a;=6,  and  y=8. 

201.  Examples  in  which  the  preceding  Rules  are  applied,  in 
the  Solution  of  Simple  Equations,  Involving  two  unknown 
Quantities. 

r.      .    r^-        .-.        «^+3     r,  .  3a;— 2y     n 

Ex.  1.  Given  2y ^=7H -^,    j       ^    ,    , 

^         4  5      '    f  to  find  the  va- 

1  .         8— y     „,,      2a;4-l  i  lues  of  a;  and  y. 
and  4a;-  -^=24 J ^,  ^  ^ 

Multiplying  the  first  equation  by  20, 

40y— 5a;— 15  =  140+ 12a;-8y; 

.".  by  transposition,  48y— 17^=155. 
Multiplying  the  second  equation  bv  6, 

24a;— 16+2y=147-6a;-3  ;  « 

.-.by  transposition,  2y+30a;=160  .  .  .  (A). 
Multiplying  this  by  24,  we  have 


SIMPLE  EQUATIONS.  153 

48y+720a?=3840; 
but48y—   17a;=    155; 


I 


.*.  by  subtraction,  737x=3685, 

and  by  division,  x=5. 

From  equation  (A),  2y=160  — 30a;; 

.-.  by  substitution,  2y=  160  —  150. 

by  division,  y=-r- ;  .*.  y=5. 

2 

The  values  of  x  and  y  might  be  found  by  any  of  the  methods 
given  in  the  preceding  part  of  this  Section  ;  but  in  solving  this 
example,  it  appears  that  Rule  I,  is  the  most  expeditious  method 
which  we  could  apply. 

Ex.  2.  Given  ^-  —-=1  —  3J!+-^, 

and  a?  :  3y  :  :  4  :  7, 

to  find  the  values  of  x  and  y. 

Reducing  the  first  equation  to  lower  terms, 

y      ^^~^_i      4+y  ,  « — y 

9  18^"  S"""*     6~' 

and  therefore,  multiplying  by  18, 

2y— 4x4- 1  =  18— 24  — 6y+3a?— 3y  ; 

.-.  by  transposition,  7=7a; — 1  ly 

But  from  the  second  equation  7a;  =  12y. 

Substituting  therefore  this  value  in  the  preceding  equation, 

it  becomes 

12y— lXy=7,  or  y=7, 

12y     84     ,^ 
and.-.ir=:   J^z=z— -=12. 
7        7 

15x4-^ 
Ex.3.Given.-?^^.l  +  -^. 

3a;-f2y       y— 5_  11j;+1,52        3y+l 
^""^  ~~6  r-        12  2"' 

to  find  the  values  of  x  and  y. 

Multiplying  the  first  equation  by  33, 

33a;— 9y+6— 3a:=33  +  15a:4-^; 

multiplying  again  by  3,  and  transposing,  we  shall  have  45ic^ 
31y=81. 

Multiplying  the  second  equation  by  12, 

6a;+4y—3y4-15=--lla; 4-152  — 18y-6  ; 

.-.  by  transposition,  19y— 5a:=131. 


154  SIMPLE  EQUATIONS. 

Multiplying  this  by  9,  171y— 45a;=1179  ; 
but  45a;— 31y=     81 ; 

.'.  by  addition,  140y=1260  ; 
and  by  division,  y=9. 
Now,  5a;=19y-131  =  171-131=40; 

/.by  division,  x=8. 

Ex.  4.  Given — -^=zl8i ^^ ,i      ^    ,   , 

15  3  7         'f  to  find  the  va- 

j  iA       6a;— 35     __  ,  .^  I  lues  of  a;  and  y. 

and  10y-| — z=55  +  10a;,       \  ^ 

o  ^ 

Multiplying  the  first  equation  by  105,  the  least  common 
multiple  of  3,  7,  and  15. 

560+21a:  =  1925— 60a?-45y4-120 ; 

.*.  by  transposition,  81a;4-45y=1485  , 
and  dividing  by  9,  9a;4-5y=165 
From  the  second  equation, 

50y-f-6a;— 35=r275+50a;, 

.'.  by  transposition,  50y — 44a;=310  ; 

and  dividing  by  2,  25y— 22a;=155  ; 

but  multiplying  the  equation  i  o5„4.45^_«25  • 

found  above,  by  5,  \  ^5y-f-45a;_-B^5  , 

.*.  by  subtraction,  67a; =670, 

and  by  division,  a;  =10. 

Now  5y=165-9a;=165-90=z:75,  .•.y=15. 

Ex.  5.  Given  i^+5|^--l,)  ,     .   ,   .  , 

^      y      y         \  to  fi^d  the  values  of  x 

,5.4     7,3       i    and  y. 
and  -+-=-+-,      ) 
a;     y     a?     2       J 

Reducing  the  first  equation  to  lower  terms, 

X    y    y 

u  ..44, 

.'.  by  transposition, =  —  1  ; 

X      y 

2     4     3 

from  the  2nd  equation,  by  transposition, 1 — =-  ; 

X     y     2 


.-.  by  addition,  -=-. 

X     2 


and,  consequently,  a; =4. 

4     4 
Now  -=—[-1=2  ;  .-.  2y=4,  and  a;=2 

y     a? 


SIMPLE  EQUATIONS.  155 

b 


a     b  \ 

-4— =n,  \ 


Ex.  6.  Given 

find  the  values  of  x  and  y. 

=  71,  \ 


Multiplying  the  first  equation  by  c,  and  the  second  by  a,  we 

shall  have 

ac  .  Jc 

=7nc, 

«     y 

.  ac     ad, 
and  — j — =na, 


by  subtraction,  {^c^adi)  .  -z=imc-'na ; 

Jc — ad 
...  y_ . 

mc — na 

.    ,  a             b  mbc—nab 

And  -=m =m— 


^c — ac? 

fnbc — mad — mbc-^-nab     nab — mad 

be — ad  ~   be — ad     * 

1     nb — md        ,        -be — ad 

•••-=1 Ty  and  x=-T -z. 

X     be — ad  no — ma 


2x 

74-— 

Ex.  7.  Given  3 -^=5-^^, 

5  3y    ' 

107 
,        44-15y    ^^y— 8- 


to  find  the  values 
of  x  and  y. 


Multiplying  the  first  equation  by  15y, 

.-.  45y-21y— 6rK=:75y-25a;— 45  ; 
and  by  transposition,  51y — 19a;=45. 
Multiplying  the  second  equation  by  2a;+5, 

2.y+5y -—^ ^-2.y-  _; 

^     ,  107     8a;4-20+30a:y4-75y 

•••^y+-r= 6^^ ' 

and  multiplying  by  6a;— 2,  tvre  shall  have 

30a;y-10y+?Hl^.Ili^=8a:+20+30a?y4-75y; 


156  SIMPLE  EQUATIONS. 

321iK-107     ^    .  „,    .  „^ 
•'• Z =8j;+85y+20, 

and  321a;— 107=32ir4-340y+80 ; 
.-.  by  transposition,  340y— 289a;=  — 187. 
The  coefficients  of  y  in  this  case,  having   aliquot  parts  ; 
multiplying  the  first  by  20,  and  the  last  by  3, 

1020y— 380a:=     900, 
and  1020y— 867a:=— 561  ; 


.*.  by  subtraction,  487a;z^l461, 

and  a;=3  ; 

consequently,  51y=45  +  19a;=45  +  57=102  ; 

.•.y=2. 

^      o   r..        o        164-60a;     16a;y-107  N 

Ex.  8.  Given  8a; ^       ,    =     /,  „ ,i      x-  j  ,-. 

3y  — 1  54-2y       f  to  find  the  va- 

,      ,  27a;2  — 12y2+38  (luesofa;andy. 

Multiplying  the  first  equation  by  5-|-2y, 

.«     .  ..           80  +  300a;  +  32y4-120a;y      _  ,^„ 

40a;4-16a;y — _^ ^^=16a;y-107  ; 

«    .  ,^«     80  +  300a;+32y+120a;y 
.-.by  transposition  40a;+107= __.^ 

and  multiplying  by  3y— 1,  we  shall  have 

120:ry— 40a:+321y— ]07=:80  +  300a:+32y+120a;y; 

.-.by  transposition,  289y— 340a;  =  187. 
And  from  the  second.equation, 

27a:2— 12y2+15a;+2y+2=r27a;2-12y2-l-38; 
.'.  by  transposition,  15a;+2y=36  ; 
whence,  the  coefficients  of  x  having  aliquot  parts,  multiplying 
the  first  equation  by  3,  and  the  second  by  68, 
867y  — 1020a;=561, 
and  136y4-1020a;=2448; 

.-.by  addition,  1003y=:3009, 

and  y=3 ; 

consequently,  15a;=36— 2y=z36-6=:30 ; 

and  .-.by  division,  a;=2. 

fir.  9.  Given  ar- 1^=20- 5£=?^,^^^  ^^^  ^^^  ^^_ 
,      .   V— 3      „^  •  73— 3y  (lues  of  a:  and  y. 

Ans.  a:=21,  and  y=20. 


SIMPLE  EQITATIONS.  167 


3^ 1 

Ex.  10.  Given h3y-4=15, 


to  find  the  values  of 


J  3y— 5  ,  o       o     ^o  V  a;  and  y. 


Ex.  11.  Given  9a:+^=70, 
5 

and  7y --=44, 


Ans.  a;=:7,  and  y=5. 

to  find  the  values  of  x  and  y. 

Ans.  a;=:6,  and  y=10. 


Ex.  12.  Given  ^  -  ?^-3y-5, 
5  4 

and  ^L_+___-18-5jc, 

to  find  the  values  of  x  and  y.  Ans.  a;=3,  and  y=2. 

X.      ,o    ^.            .  ,       3y+4a;     „      9y4-33 
Ex.  13.  Given  a: 4-1 ^ — =7 ^TJ-"' 

„      5a;— 4y             lly— 19 
and  y-3 2"^=^ 4 ' 

to  find  the  values  of  x  and  y.  Ans.  a:=6,  and  r/=5. 

Ex.  14.  Given  4a:4-i^-=2y4-5+^^^^^, 
4  lb 


to  find  the  values  of  x  and  y.  Ans.  a;=3,  and  y=4. 

Ex.  15.  Given  x-5^+17=5y+lf±I, 

22— 6y       5x-7     a;+l       8y+5 

and  -^ rr =-6 18~' 

to  find  the  values  of  x  and  y.  Ans.  a?=:8,  and  y=2. 

Ex.  16.  Given — ^- — =4H — , 

6^3  2      ' 

,  2a;+y       9a;  — 7_3y+9       4a;+5y 
__  ______  16     » 

to  find  the  values  of  x  and  y.  Ans.  a;=9,  and  y=4. 

„      ,„    r.-        7a;4-6  ,  4y— 9     „         13— a;       3y— a? 

Ex.  17.  Given  -— f — {—^ — =3a; ^ — ,and 

113  2  o 

3a?+4  :  2y— 3  :  :  5  :  3,  to  find  the  values  of  a;  and  y. 

Ans.  a;=7,  and  y=9. 
15 


158  SIMPLE  EQUATIONS. 

Ex.  18.  Given — ^ ^— ^ =9+ ^ ,and 

2d  o 

fjL_  :  -^ U4a?  :  :  4  :  21,  to  find  the  values  of  x  and  y. 

3  4  ^ 

Ans.  a=5,  and  y=4. 

Ex.  19.  Given —-^^^-^^^ 1^—-^  =  5+^--,    and 

10  .  15  5 

9y+5a?— 8       jr-f  y     7a!-f  6  ,    ^    ,  ^-u        i  r         j 

-^ ^= — -— ,  to  find  the  values  oi  x  and  y. 

12  4  11    '  ^ 

Ans.  Of =7,  and  y=9. 

Ex.  20.  Given  3a;—2y=:  15,  >  .^  a  ♦!,        i  r  a 

J      ,  ,  ^  1  K       n'     o   ^  tohnd  the  values  of  x  and  y. 

and  y+iO  ^  «— 15  : :  7  :  3,  >  ^ 

Ans.  af=i45,  and  y=60. 

Ex.  21.  Given  aj+lSQ  :  y— 50  : :  3  :  2,  >  to   find  the  va- 

and  a?— 50  :  y+100  :  :  5  :  9,  >  lues  of  a?  and  y. 

Ans.  a;  =  300,  and  y=:350. 

Ex.  22.  Given  (a:+5).  (y+7)=(a;+l){y—9)  +  112, 

and  2ic4-10  =  3y+l,  to  find  the  values  of  x  and  y. 

Ans.  x—'^j  and  y=5. 

2a;  — 4y+3 
151— 16j:     9a:y— 110 

to  find  the  values  of  a;  and  y.  Ans.  a; =9,  and  y=2. 

,^     .  ^       ,      128«2_i8ya-f217 
Ex.24.   Gwenl6.+6y-l^        8.-3y%2— - 

10a:4-lQy-35_  54 

^  2a;-f-2y-i-3  "~"         3a;+2y-r 

to  find  the  values  of  x  and  y.  Ans.  x—^,  and  y=5. 

\  III.    RESOLUTION  OF  SIMPLE  EQUATIONS, 

Involving  three  or  more  unknown  Quantities. 

202.  When  there  are  three  independent  simple  equations 
involving  three  unknown  quantities. 

RULET, 

From  two  of  the  equations,  find  a  third,  which  involves  only 
two  of  the  unknown  quantities,  by  any  of  the  rules  in  the  pre- 
ceding Section  ;  and  in  like  manner  from  the  remaining  equa- 
Uon,  and  one  of  the  others,  another  equation  which  contains  the 


SIMPLE  EQUATIONS.  159 

same  two  unknown  quantities  may  be  deduced.  Having 
therefore  two  equations,  which  involve  only  two  unknown 
quantities,  these  may  be  determined  ;  and,  by  substituting 
their  values  in  any  of  the  original  equations,  that  of  the  third 
quantity  will  be  obtained- 

203.  If  there  be  four  unknown  quantities,  their  values  may 
be  found  from  four  independent  equations-  For  from  the  four 
given  equations,  by  the  rules  in  the  last  Section,  three  may  be 
deduced  which  involve  only  three  unknown  quantities,  the  va- 
lues of  which  may  be  found  by  the  last  Article  ;  and  hence  the 
fourth  may  be  found  by  substituting  in  any  of  the  four  given 
equations,  the  values  of  the  three  quantities  determined. 

If  there  be  n  unknown  quantities,  and  n  independent  equa 
tions,  the  values  of  those  quantities  may  be  found  in  a  similar 
manner-  For  from  the  n  given  equations,  n—\  may  be  de- 
duced, involving  only  n  —  I  unknown  quantities  ;  and  from 
these  n  — 1,  n— 2  may  be  obtained,  involving  only  n—2  un- 
known quantities  ;  and  so  on,  till  only  one  equation  remains, 
involving  one  unknov/n  quantity  ;  which  being  found,  the  va- 
lues of  all  the  rest  may  be  determined  by  substitution. 

Ex.  1.    Given  x-^y^z=:29,      \ 

x-|-2y-i- 35^=62,  f  to  find  the  values  of  a:,  y, 
X  ,  y  ,  z     ,  ^       C  and  z. 

Subtracting  the  first  equation  from  the  second, 

y-l-2^=^33     .     .     .     (A). 
Multiplying  the  thircL^quation  by  12,  the  least  common 
multiple  of  2,  3,  and  4, 

6a; -l-4y-f  3:^=120 
multiplying  the  I  st  equation  by  6,  6a;  -|-  6y  -f-  6^^  =  1 74  ; 

.-.  by  subtraction,  2i/-\-3z=54  ; 
but,  multiplying  equation  (A)  by  2,     2^4-4^=66  ; 

.*.  by  subtraotion,  ;s-=:12- 
From  equation  (A),  by  transposition,  y  =  33— 2,^  ; 
.-.  by  substitution,  y=33— 24,  or  y  =  9. 
From  the  first  equation,  by  transposition, 
a;=:29—y—z; 

.-.  by  substitution,  a:=29— 9  — 12, 
and  a:i:=29— 21,  .■.x=8. 
In  like  manner,  had  the  first  equation  been  multiplied  by  2, 
and  subtracted  from  the  second,  an  equation  would  have  re- 
sulted, involving  only  x  and  z  ;  and  had  it  been  multiplied  by 
4,  and  subtracted  from  the  third  when  cleared  of  fractions. 


160  SIMPLE  EQUATIONS. 

another  equation  would  have  been  obtained,  involving  also  x 
and  z  ;  whence  by  the  preceding  rules,  the  values  of  x  and  z 
could  be  found,  and  consequently  the  value  of  y  also,  by  sub- 
stitution. 

Or  if  the  first  equation  be  multiplied  by  3,  and  the  second 
subtracted  from  it,  an  equation  would  arise,  involving  only  x 
and  y  ;  and  if  the  first  when  multiplied  by  3,  be  subtracted 
from  the  third  when  cleared  of  fractions,  another  would  arise 
involving  only  x  and  y  ;  whence  the  values  of  x  and  y  might  be 
determined.     And  hence  the  third,  that  of  ^,  might  be  found. 


SECOND  METHOD. 

From  the  first  equation,  x=29—y—z; 
substituting  this  value  of  x  in  the  second  equation, 
29— y— 2+2^+3^=62  ; 

.-.by  transposition,  y=33—2z. 
Also  substituting,  in  the  third  equation,  the  value  of  x  found 
from  the  first, 

29-y-z     y     z 

2         ^3^4~   ^ 
multiplying  this  equation  by  12,  the  least  common  multiple  (rf 
2,  3,  and  4, 

174--'6y—ez-^4y+3z=120, 

and  by  transposition,  2y-{-3z=z54: ; 
in  which,  substituting  the  value  of  ^ound  above, 

2(33-2;^)+32=54  ; 
or  66— 4^+3;^=:54  ; 
.*.  by  transposition,  2:= 12  ; 
whence  y=33— 2^  =  33— 24  =  9, 
and  a;=.29— y— 2=29  — 9  — 12  =  8. 
It  may  be  observed,  that  there  will  be  the  same  variety  of 
solution,  as  in  the  last  case  according  as  x,  y,  or  2,  is  exter- 
minated. 

THIRD  METHOD. 

The  values  of  x,  found  in  each  of  the  equations,  being 
compared,  will  furnish  two  equations  each  involving  only  y 
and  z  ;  from  which  the  values  of  y  and  z  may  be  deduced  by 
any  of  the  rules  in  the  preceding  Section,  and  hence,  the  va- 
lue of  X  can  be  readily  ascertained. 


SIMPLE  EQUATIONS.  161 

The  same  observation  applies  to  this  method  of  solution,  as 
did  to  the  last. 

In  some  particular  equations,  two  unknown  quantities  may 
be  eliminated,  at  once. 

Ex.  2.    Given  x+y-{-z=3l  _ 

'  to  find  the  values  of  a;,  y,  &  z. 


x+y-{-z=3l  \ 
x+i/—zz=25  > 
X — y—z=:9    } 


Adding  the  first  and  third  equations,  2a;=40  ; 

.♦.  a?=20. 
Subtracting  the  second  from  the  first,  2z=6; 

,:z=3; 
and  subtracting  the  third  from  the  second, 

2y=16;  .-.3^=8. 
Ex.  3.    Given  ^  x—z=3,  J  to  find  a:,  y,  and  z. 


rx-y=2,) 

^  X — Z=:3,  >    to 

(y-^  =  l,) 


Here  subtracting  the  first  equation  from  the  second,  we 
have  t/—z  —  l  ;  which  is  identically  the  third. 

Therefore,  the  third  equation  furnishes  no  new  condition  ; 
but  what  is  already  contained  in  the  other  two ;  and,  conse- 
quently, the  proposed  equations  are  indeterminate  ;  or,  what 
is  the  same,  we  may  obtain  an  infinite  number  of  values  which 
•will  satisfy  the  conditions  proposed. 

204.  It  is  proper  to  remark,  that  in  particular  cases.  Ana- 
lysts make  use  of  various  other  methods  besides  those  pointed 
out  in  the  practical  rules  ;  in  the  resolution  of  equations, 
which  greatly  facilitate  the  calculation,  and  by  means  of 
which,  some  equations  of  a  degree  superior  to  the  first,  may 
be  easily  resolved,  after  the  same  manner  as  simple  equations. 

We  shall  illustrate  a  few  of  those  artifices  by  the  following 
examples. 

Ex.  4    Given  -+-=1,    1 
X     y     8       \ 

-+-=-,     J>  to  find  the  values  of  a?,  y,  and  s. 

X       Z       i) 

and— 1— =— , 
y    z     10 

15* 


162  SIMPLE  EQUATIONS. 


By  adding  the  three  equations,  we  i 

x^y^z     8^9^10 
Or,  dividing  by  2, 

xy^z     720 

shall  have 

121 
•■~360' 

From  this  subtracting 
we  shall  have 

1_  31 

;^~720' 

each  of  the  three  first  equations,  and 

720              ^„  7 
or  .=:--;.. ..==23-; 

1      41 
y~720' 

720 

— '!f 

1       49 
a;~720' 

720 
"'  ="=  49   ■'  • 

■■'< 

Ex.  5.  Given  2a;=:y4-;sr-|-w,^ 

di/=x-{-z-\-u,  f    to  find  the  values  of  x,  y,  z, 

4:ZT=zx-\-y-\-u,  I   and  u. 

and    u=zx — 14,      ) 

By  adding  x  to  each  member  of  the  first  equation,  y  to  the 

second,  and  z  to  the  third,  we  shall  get 

x-^y-\-z-\-u=z'dx=.Ayz=:^z  \ 

3a;  3a; 

and  from  thence,  ^--r^  and  y=r— - ; 

which  values  being  substituted  in  the  first  equation,  we  have 
3a?  .  3a;  .  13a; 

but,  by  the  fourth,  equation,  w = a; — 1 4  ; 

13a; 
.♦.a;— 14=— -,  or  20a;— 280=13ap; 

—  3a; 

whence  a;=40  :  consequently  y=— =  30,  ;?=24,  and  «=» 

—14=26. 

Ex.  6.  Given  4a;  — 4v—43:=24,  "i  ,    ^    i  ^r        ^         c 

c       o       o       o  ^    Mo  find  the  values  of  x.  y, 
6y— 2a;— 2;?=24,  >       i  '^' 

and  7z--  y—  x=24,  J 
By  putting  x-\-y-{-z^S,  the  proposed  equations  become 
8a;— 4S  =  24,  8y-2S=24,  8;^-S=24  ; 

.-.  a;=3  +  iS,  y==3+iS,  ;?=3+ JS. 
By  adding  these  three  equations,  we  have 

a;+y+;^=9+|S  ;  whence  S=72. 


SIMPLE  EQUATIONS.  163 

Substituting  this  value  for  S,  in  a;,  y,  and  jsr,  we  shall  find 

a'=39,  y— 21,  and  z=12. 

Ex.7.  Given        oc-\- y-\-z =90,  \-      /.    ,  ,,         ■,   ■      e 

2a:+40i3y+20;  i  '^  ^""^  '^^  ^^^^^^  °^  ^'  ^^ 
and  2x-4z+40=^l0,  )  ^^"^  ^• 

Ans.  x=35,  y=30,  and  z=25. 


Ex.  8.  Given  a?4-<z=  V+^j    ^  *    x:  j  *i,         i  r 

o       o     '  to  find  the  values  of  x,  y, 
y+a=2x+2z,^^^^^^  >y 


)  to 

and  z-\-a=Zx-\-^y,  )  ^" 

a  5ct        ,        7a 

Ans.  a;=— '  ^^JY'  ^"^^  ^=  n 

Ex.  9.    It  is  required  to  find  the  values  of  x,  y,  and  Zf  in 
the  following  equations ; 

a;+y=13,  x-{-z=14j  and  i/-\-z=15. 

Ans.  x=6,  y=7,  and  z=8, 

Ex.  10.  In  the  following  it  is  required  to  find  the  values  of 
X,  y,  and  z. 


X    y  ,  z 
3+1+5=94, 

X  ,  y     z 


r  a?=48, 
n-=120, 

(  ;S=240. 


Ex.11.  Given  a:+y+;sr=26,  ^  ,    c  a   .x.         i  r 

^  ^   f  to  find  the  values  of  x,  y. 

and  X — z         =  6,  ^ 

Ans.  a;=12,  y  =  8,  and  z=zQ. 

Ex.  12.  Given  0?+  y4-  ;3-=  9,  )  ,    ^    ,  ,,        ,  . 

^  '  '  to  find  the  values  of  a?,  y, 


a;+  y+  ;3-=   y,  ^ 
a;+2y+3^=16,  >       , 
and  ^4-  y-2^=   3^^"^^ 


Ex.  13.  Given  a;+  y+  z=l2,  \  ,    .    .  ,.         ^         c 

a,+2v+3;^=20  J  ^^  ?"^  ^^^  ^*^"®^  ^'^  ^'  y» 


Ans.  a:=4,  y=3,  and  z=l2. 
=  12 
=20 
and  \oc-\-\y-\-   z- 

Ans.  a;=6,  y=4,  and  zz=z2. 

Ex.  14.  Given  a;+y— ^=8,  a;+;^— y=:9,  and  y-\-z—x=^ 
10  ;  to  find  the  values  of  x,  y,  and  z. 

Ans.  xz=.%\,  y  =  9,  and  ;^=9^. 
Ex.  15.    Given   a;+^y  =  100,  y4.i;j=lOO,   and  ;?+Ja;= 
100  ;  to  find  the  values  of  a;,  y,  and  z. 

Ans.  a?=:64,  y=72,  and  ;y=84. 


164  SOLUTION  OF  PROBLEMS 

Ex.  16.  Given  a:4-iy=357,y-|-i^=476,  if  4-iM=595,  and 
u-\-^x=z7l4j  to  find  the  values  of  cc,  y,  z,  and  u. 

Ans.  a:=190,  y  =  334,  z=A26,  and  w  =  676. 


§  IV.    SOLUTION  OF  PROBLEMS  PRODUCING  SIMPLE  EQUATIONS, 

Involving  more  than  one  unknown  Quantity. 

205.  The  usual  method  of  solving  determinate  problems  of 
the  first  degree,  is,  to  assume  as  many  unknown  letters,  name- 
ly, X,  y,  z,  &c.,  as  there  are  unknown  numbers  to  be  found  ; 
then,  having  properly  examined  the  meaning  and  conditions  of 
the  problem,  translate  the  several  conditions  into  as  many 
distinct  algebraic  equations  ;  and,  finally,  by  the  resolution  of 
these  equations  according  to  the  rules  laid  down  in  Chapter 
IV,  the  quantities  sought  will  be  determined.  It  is  proper  to 
observe  that,  in  certain  cases,  other  methods  of  proceeding 
may  be  used,  which  practice  and  observation  alone  can  sug- 


Problem  I. 

There  are  two  numbers,  such,  that  three  times  the  greater 
added  to  one-third  the  lesser  is  equal  36  ;  and  if  twice  the 
greater  be  subtracted  from  6  times  the  lesser,  and  the  remain- 
der divided  by  8,  the  quotient  will  be  4.  What  are  the  num- 
bers ? 

Let  X  designate  the  greater  number,  and  y  the  lesser  num- 
ber. 

Then  3a:-f^=36;)  ^^    ,  ,n«/A\ 

'3  f  .    <  9a;+  y  =  108  (A), 

.  6y— 2a;     ,     (  ''  \  Gy-2x=   32  (B) ; 

and  -^— — =4  ;  \ 

o  y 

Multiplying  equation  (A)  by  6,  Oy-}- 54a:  =  648  ; 
but  6y—  2x=   32  ; 

.-.  by  subtraction,  56a:rr:616, 
and  by  division,  a:=ll. 
From  equation  (A),  y=108— 9a? ; 

.-.  by  substitution,  y= 108—99,  or  y— 9. 


PRODUCING  SIMPLE  EQUATIONS.         165 

Prob.  2.  After  A  had  won  four  shillings  of  B,  he  had  only- 
half  as  many  shillings  as  B  had  left.  But  had  B  won  six  shil- 
lings of  A,  then  he  would  have  three  times  as  many  as  A 
would  have  had  left.     How  many  had  each  1 

Let  x=  designate  the  number  of  shillings  A  had,  and  y= 
the  number  B  had ; 

then  y— 4=:2a?+8, 
and  ■i/-\-6=z3x—l8', 

.-.  by  subtraction,  10=a?— 26, 
and  by  transposition,  36  =  x,  or  a; =36 
by  substitution,  y+6=3x  36  —  18 
and  by  transposition,  y  =  84 
.-.  A  had  36,  and  B  84. 

Prob.  3.  What  fraction  is  that,  to  the  numerator  of  which 
if  4  be  added,  the  value  is  one-half,  but  if  7  be  added  to  the 
denominator,  its  value  is  one-fifth  ? 

Let  ^=  its  numerator,  )  ,^^„  ^^^  f^^^^j^^  x 
yz=z  denommator,    )  y 

Add  4  to  the  numerator,  then =i,  .-.  2a;+8=y  ; 

y 

Add  7  to  the  denominator,  then  — — =J,  .-.  5a?=y+7; 

by  subtraction,  3x— 8=7  ; 

by  transposition,  3a;=:15  ;  .".  xz^b  ; 

and  y=2a?+8  ;  .-.by  substitution,  y=10-f-8=18, 

5 
and  the  fraction  is  -— . 
18 

Prob.  4.  A  and  B  have  certain  sums  of  money,  says  A  to 
B,  give  me  15/  of  your  money,  and  I  shall  have  5  times  as 
much  as  you  have  left :  says  B  to  A,  give  me  5/  of  your 
money,  and  I  shall  have  exactly  as  much  as  you  will  have 
left.     What  sum  of  money  had  each  ? 

Let  x=i  A's  money,  >  then  x-\-\b=.  what  A   would  have, 
y=z  B's,  S  after  receiving  15Z  from  B. 

y  — 15=  what  B  would  have  left. 
Again,y-f  5=  what  B  would  have  after  receiving  5/ from  A. 
x  —  b—  what  A  would  have  left. 
Hence,  by  the  problem,  a;+15=5  x(y— 15)  =  5y--75, 
and  y+5=a?— 5. 


166  SOLUTION  OF  PROBLEMS 

by  transposition,  5y— a?=i90, 
and  y — a;z=  — 10  ; 

/.  by  subtraction,  4y=100, 
and  by  division,  y=25  B's  money. 
From  the  second  equation,  a:=y-f  10  ; 

.'.  by  substitution,  a;=25  4-10=35  A's  money. 

Prob.  6.  A  person  was  desirous  of  relieving  a  certain  num- 
ber of  beggars  by  giving  them  2s.  6d.  each,  but  found  that  he 
had  not  money  enough  in  his  pocket  by  3  shillings ;  he  then 
gave  them  2  shillings  each,  and  had  four  shillings  to  spare. 
What  money  had  he  in  his  pocket ;  and  how  many  beggars 
did  he  relieve  ? 

Let  x=.  money  in  his  pocket  {in  shillings)  ; 
t/z=  the  number  of  beggars. 

Then  2^  Xy,  or  -^=  number  of  shillings  which,  would  have 

been  given  at  2s.  6d.  each ; 
and  2  Xy,  or  2y=:     ....     at  2s.  each. 

Hence,  by  the  problem, -^=:a?-j- 3(A), 

and  2yz=a?— 4(B). 

V 
.-.  by  subtraction,  ^=7, 
lit 

or  y=14,  the  number  of  beggars. 

From  equation  (B),  a;=2y4-4=2  x  14-^-4,  by  substitution, 

.-.  a;zr:32,  the  shillings  in  his  pocket. 

Prob.  6.  There  is  a  certain  number,  consisting  of  two  digits. 
The  sum  of  those  digits  is  5  ;  and  if  9  be  added  to  the  number 
itself,  the  digits  will  be  inverted.     What  is  the  number  ? 

Here  it  may  be  observed,  that  every  number  consisting  of 
two  digits  is  equal  to  10  times  the  ;ligit  in  the  tens  place,  plus 
that  in  the  units  ;  thus,  24=^2  X  104-4x=20-f  4. 

Let  x=  digit  in  the  units  place  ; 
y=.  that  in  the  tens. 

Then  10a;-4-y=  the  number  itself, 
and  10y-f-a;=  the  number  with  its  digits  inverted. 

Hence,  by  the  problem,  a:-|-y=5(A), 
and  10a;-|-y+9=:10y4-^»  or  by  transposition,  9a;— 9y=— 9  ; 
.'.by  division,  a;— y  =  —  1(B). 

Subtracting  equation  (B)  from  (A),  2y=sQ  ; 


PRODUCING  SIMPLE  EQUATIONS.  167 

.•.y=3,  and  a:=5— y=5— 3=2  ; 
.-.  the  number  is  (10a;+y)=23. 
Add  9  to  this  number,  and  it  becomes  32,  which  is  the  num- 
ber with  the  digits  inverted. 

pROB.  7.  A  sum  of  money  was  divided  equally  amongst  a 
certain  number  of  persons  ;  had  there  been  four  more,  each 
would  have  received  one  shilling  less,  and  had  there  been  four 
fewer,  each  would  have  received  two  shillings  more  than  he 
did  :  required  the  number  of  persons,  and  what  each  received. 

Let  X  designate  the  number  of  persons,  • 

y  the  sum  each  received  in  shilUings  ; 
then  xy  is  the  sum  divided  ; 

.^^%t.%l%-l^)lZ\\     by  the  question; 

.•.xy-\-A:y — X — 4  =  a?y,  or  4y —  a:=4, 

and  xy—Ay-^2x—S=ixy,  or  — 4y-|-2j:  =  8  ; 

.*.  by  addition,  a?=12  ; 

and  4y=44-a?=4  +  12 ;  .'.^=4. 

Prob.  8.  A  man,  his  wife,  and  son's  years  make  96,  of 
which  the  father  and  son's  equal  the  wife's  and  15  years  over, 
and  the  wife  and  son's  equal  the  man's  and  two  years  over. 
What  was  the  age  of  each  ? 

Suppose  X,  y,  and  z  =:  their  respective  ages. 
1st  condition  x-{-y-\-z=:96,  ^ 
2nd    .    .    .    x-\-z=y-^l5,  >      by  the  problem. 
3d       ...     y-\-z=x-\-  2,) 

Subtracting  the  2nd  from  the  1st,  y=96— y— 15  ; 

.'.2y=z81,  and  y  —  40J  by  division. 

Subtracting  the  3d  from  the  1st,  x=96—x—2  ; 

.•.by  transposition  and  division,  x=47, 

And  from  the  1st,  z=z96—y—x  ;  .-.  ^^=8^. 

And  their  ages  are  47,  40J,  and  8J  respectively. 

Prob.  9.  A  labourer  working  for  a  gentleman  during  12 
days,  and  having  had  with  him,  the  first  seven  days,  his  wife 
and  son,  received  74  shillings  ;  he  wrought  afterwards  8  other 
days,  during  5  of  which  he  had  with  him  his  wife  and  son,  and 
he  received  50  shillings.  Required  the  gain  of  the  labourer 
per  day,  and  also  that  of  his  wife  and  son. 

Let  x=  the  daily  gain  of  the  husband, 
yzz:  that  of  the  wife  and  son  ; 
12  days  work  of  the  husband  would  produce  12a:, 
7  of  the  wife  and  son  would  be  7y  ; 


168  SOLUTION  OF  PROBLEMS 

.-.by  the  first  condition,  12a? -f  ly=  74 

and  by  the  second,    8a;-|-  ^y—  50 

Multiplying  the  1st  equation  by  2,  24a;4-14y=rl48 

2nd     .     .     by  3,  24x4"  15y= 150 


/.by  subtraction,  y=2. 

And  from  the  2nd,  8a;=50— 5y=50  — 10  ; 

.'.by  division  a:=5. 

Consequently  the  husband  would  have  gained  alone  5s.  per 
day,  and  the  wife  and  son  2  shillings  in  the  same  time. 

506.  Let  us  now  suppose  that  the  first  sum  received  by  the 
workman  was  46s,  and  the  second  30s,  the  other  circumstances 
remaining  the  same  as  before  ; 

The  equations  of  the  question  would  be 

12a;+7y=46,  and  8a;-f-5y=30. 

From  whence  we  find,  by  proceeding  as  above, 
a:=5,  and  y  =  — 2. 

By  putting  in  the  place  of  a?  its  value  5,  in  the  above  equa- 
tions, they  become 

604-7^=46,  and  404-53/  =  30. 

The  inspection  alone  of  these  equations  show  an  absurdity. 
In  fact,  it  is  impossible  to  form  46  by  adding-  an  absolute  num- 
ber to  60,  which  is  already  greater  than  it,  and  in  like  man- 
ner it  is  impossible  to  form  30  by  adding  an  absolute  number 
to  40. 

Consequently  what  we  attributed  as  a  gain  to  the  labour  of 
the  wife  and  son,  must  be  an  expense  to  the  husband,  which 
is  also  verified  by  the  result  y=—2. 

207.  The  negative  value  of  y  makes  known  therefore  a 
rectification  in  the  enunciation  of  the  problem  ;  since  that,  in- 
stead of  adding  7y  to  12a:  in  the  first  equation,  and  5y  to  8a:  in 
the  second,  y  being  considered  a  positive  or  an  absolute  num- 
ber, we  must  subtract  them  in  order  to  have  the  sum  given  for 
the  common  wages  of  these  three  persons  ;  or,  what  is  the 
same  thing,  if,  in  place  of  considering  the  money  attributed  to 
the  wife  and  son  as  a  gain,  we  would  regard  it  as  an  expense 
made  by  them  to  the  charge  of  the  workman  ;  then  we  must 
subtract  this  money  from  what  the  man  would  have  gained 
alone,  and  there  would  be  no  contradiction  in  the  equations, 
since  they  would  become 

60-7y  =  46,  and  40-5y=30  ; 
from  either  of  which  we  would  derive  y=2  ;  and  we  should 
therefore  conclude  that  if  the  workman  gained  5s.  per  day,  his 
wife  and  son's  expense  is  2s.,  which  can  be  otherwise  verified 
thus : 


PRODUCING  SIMPLE  EQUATIONS.         169 

For  12  days  work,  he  receives  5  x  12  or  60s. ;  the  expense 
of  his  wife  and  son  for  7  days,  is  2  X  7  or  14s. ;  and  there  re- 
mains 46  shillings. 

Again,  he  receives  for  8  days  work  5  X  8  or  40s.  ;  the  ex- 
pense of  his  wife  and  son  during  5  days,  is  2x5  or  10s. ; 
therefore  his  clear  gain  is  30  shillings. 

208.  It  is  very  evident  that,  in  place  of  the  enunciation-of 
(Prob.  9),  we  must  substitute  the  following,  in  order  that  the 
problem  proposed  may  be  possible,  with  the  above  given 
quantities  : 

A  labourer  working  for  a  gentleman  during  12  daySy  having 
had  with  him,  the  first  7  days,  his  wife  and  son,  who  occasion  an 
expense  to  him,  received  46  shillings  ;  he  has  wrought,  after- 
wards, for  8  other  days,  on  5  of  which  he  had  with  him  his  wife 
and  son,  whose  expenses  he  must  still  defray,  and  he  received 
30  shillings.  Required  the  salary  of  the  workman  per  day,  and 
also  the  expense  of  his  wife  and  son  in  the  same  time. 

Designating  by  x  the  daily  wages  of  the  workman,  and  by 
y  the  expense  of  his  wife  and  son,  for  the  same  time  ;  the 
equations  of  the  problem  shall  be 

\2x—ly  —  AQ,  and  8a;— 5y  =  30  ; 
which,  being  resolved,  will  give 

a;  =  5s.,  and  yz=2s. 

209.  Although  negative  values  do  not  answer  the  enuncia- 
tion of  a  concrete  question,  as  has  been  observed  (Art.  174), 
yet  they  satisfy  the  equations  of  the  problem,  as  may  be  rea- 
dily verified,  by  substituting  5  for  x,  and  —2  for  y,  in  the 
equations  (Art.  206),  since  they  would  then  become  identi- 
cally equal. 

Prob.  10.  Two  pipes,  the^vater  flowing  in  each  uniformly, 
filled  a  cistern  containing  330  gallons,  the  one  running  during 
5  hours,  and  the  other  during  4  ;  the  same  two  pipes,  the  first 
running  during  two  hours,  and  the  second  three,  filled  another 
cistern  containing  195  gallons.  The  discharge  of  each  pipe 
is  required. 

Let  X  represent  the  discharge  of  the  first  in  an  hour  ;  y  that 
of  the  second  in  the  same  time. 

And  in  order  to  have  a  general  solution,  put  a=zb,  5=4, 
c=330,  a'=2,  5'  — 3,  c'  =  195  ;  then  by  the  conditions  of  the 
problem  we  shall  have  these  two  equations, 

ax-\-hy=c,  and  a^x-\-h'y=zc'  ; 
which,  being  resolved  (Art.  190),  will  give 
h'c — hc^        -  ac' — a'c 

aa^ab  ^     al/—ao 

16 


170  SOLUTION  OF  PROBLEMS 

Now,  by  restoring  the  values  of  a,  i,  c,  cScc,  we  have 
990—780     210 

975-660      ,^ 

.Thus,  the  first  pipe  discharges  30  gallons  per  hour,  and  the 
second  45. 

210.  Let  us  now  suppose  that  the  first  pipe  running  during 
3  hours,  and  the  second  during  7,  filled  a  cistern  containing 
190  gallons  ;  that  afterwards,  the  first  running  4  hours,  and 
the  second  6,  filled  a  cistern  containing  120  gallons. 

In  this  case,  a=3,  5=: 7,  c=190,  a'  —  i,  b'  =  6,  c'z=:120; 
and,  consequently,  &'c—^c'=  1140  — 840  =  300,  ab'^a'b=l8 
—28=:  — 10,  ac'-a^c=360— 760zz:-400,  which  will  give  x 
=  —30,  andy  =  40. 

In  order  to  understand  the  meaning  of  these  results,  we 
must  return  again  to  the  conditions  of  the  problem,  or,  what 
amounts  to  the  same,  we  must  try  how  these  values  of  x  and 
y  satisfy  the  equations  of  the  problem  : 

Thus,  if  we  substitute  —30  for  x,  and  40  for  y,  in  the 
equations  3a;4-7y=:190  and  4a;-+-6y  =  120,  resulting  from  the 
above  problem,  we  find  first,  that  3a:=  — 90,  and  7y  =  280, 
consequently  3a;+7y=  — 90  +  280,  which  in  efiect  is  equal 
to  190.  In  like  manner  4a;+ 6y  is  found  to  be  — 120+240, 
which  is  equal  to  120. 

Having,  therefore,  discovered  how  the  values  — 30  and  +40 
of  X  and  y  answer  the  equations  3ir+7y  =  190  and  4a;  +  6y  = 
120,  we  perceive  at  the  same  time  how  they  would  answer  the 
conditions  of  the  problem  ;  for*since  the  use  that  has  been 
made  of  the  quantities  3a;  and  4.x,  which  express  the  quanti- 
ties of  water  discharged  by  the  first  pipe  in  the  first  and  se- 
cond operation,  was  to  subtract  them  from  7y  and  from  6y, 
which  express  the  quantities  furnished  in  the  same  operations 
by  the  second  pipe.  The  first  pipe  must  be  considered  in 
this  case  as  depriving  the  cisterns  of  water  instead  of  fur- 
nishing any,  as  it  did  in  the  preceding  problem,  and  as  it  was 
supposed  in  expressing  the  conditions  of  this  problem. 

211.  Hence,  in  almost  every  question  solved  after  a  gene- 
ral manner,  we  may  always  conclude  that  when  the  value  of 
the  unknown  quantity  becomes  negative,  the  quantity  ex- 
pressed by  it  should  be  considered  as  being  of  an  opposite 
kind  from  what  it  was  supposed  in  expressing  the  conditions 
of  the  problem. 

What  has  been  said  with  respect  to  unki\pwn  quantities,  is 


PRODUCING  SIMPLE  EQUATIONS.    #  171 

equally  applicable  to  known  quantities,  that  is,  when  a  gene- 
ral solution  is  applied  to  any  particular  case,  if  any  of  the  gi- 
ven quantities  a,  6,  c,  &c.  in  the  problem,  are  negative. 

212.  Let  it  be  proposed,  for  example,  to  find  what  should 
be,  in  the  foregoing  problem,  the  discharges  of  two  pipes, 
that  the  first  furnishing  water  during  3  hours,  and  the  second 
4,  may  fill  a  cistern  containing  320  gallons,  and  that  the  se- 
cond pipe  afterwards  furnishing  water  during  6  hours,  whilst 
the  first  discharges  it  during  3  hours,  may  fill  a  cistern  con- 
taining 180  gallons. 

We  have  only  to  put  in  the  general  solution  (Art.  209), 
a— 3,  b  —  4,  c=320,  a'  =  — 3,  b'  =  6,  c'=zl8(J,  and  there  will 
result  a;=40,  and  y=:50. 

From  whence  it  appears  that  the  discharge  of  the  first  j)ipe 
is  40  gallons  per  hour,  either  to  carry  away  the  water  as  in 
the  second  operation,  or  to  furnish  it  as  in  the  first,  and  the 
discharge  of  the  second, *50  gallons  an  hour,  which  it  furnishes 
in  both  operations. 

Prob.  11.  a  certain  sum  of  money  put  out  to  interest, 
amounts  in  8  months  to  297/.  12.y.  ;  and  in  15  months  its 
amount  is  306Z.  at  simple  inter^t.  What  is  the  sum  and  the 
rate  per  cent  ?  Ans.  288/.  at  5  per  cent. 

Prob.  12.  There  is  a  number  consisting  of  two  digits,  the 
second  of  which  is  greater  than  the  first,  and  if  the  number 
be  divided  by  the  sum  of  it«  digits,  the  quotient  is  4 ;  but  if 
the  digits  be  inverted,  and  that  number  divided  by  a  number 
greater  by  2  than  the  difference  of  the  digits,  the  quotient  be- 
comes 14.     Required  the  number.  Ans.  48. 

Prob.  13.  What  fraction  is  that,  whose  numerator  being 
doubled,  and  denominator  increased  by  7,  the  value  becomes 
I ;  but  the  denominator  being  doubled,  and  the  numerator 
increased  by  2,  the  value  becomes  |-  ?  Ans.  ^. 

Prob.  14.  A  farmer  parting  with  his  stock,  sells  to  one 
person  9  horses  and  7  cows  for  300  dollars  :  and  to  another, 
at  the  same  prices,  6  horses  and  13  cows  for  the  same  sum. 
What  was  the  price  of  each  ? 

Ans.  the  price  of  a  cow  was  12  dollars,  and  of  a  horse  24 
dollars. 

Prob.  15.  A  Vintner  has  two  casks  of  wine,  from  the  great- 
er of  which  he  draws  15  gallons,  and  from  the  less  11  ;  and 
finds  the  quantities  remaining  in  the  proportion  of  8  to  3.  Af- 
ter they  became  half  empty,  he  puts  10  gallons  of  water  into 
each,  and  finds  that  the  quantities  of  liquor  now  in  them  are 
as  9  to  5.     How  many  gallons  will  each  hold  ? 

Ans.  the  larger  79,  and  the  sm^l^^r  ,35  gallons 


\  OF     ■. K     J^ 


172  0  SOLUTION  OF  PROBLEMS 

Frob:  16.  A  person  having  laid  out  a  rectangular  bowling- 
green,  observed  that  if  each  side  had  been  4  yards  longer,  the 
adjacent  sides  would  have  been  in  the  ratio  of  5  to  4  ;  but  if 
each  had  been  4  yards  shorter,  the  ratio  would  have  been  4 
to  3.     What  are  the  lengths  of  the  sides  ? 

Ans.  36,  and  28  yards. 

Prob.  17.  a  sets  out  express  from  C  towards  D,  and  three 
hours  afterwards  B  sets  out  from  D  towards  C,  travelling  2 
miles  an  hour  more  than  A.  When  they  meet  it  appears  that 
the  distances  they  have  travelled  are  in  the  proportion  of  13 
to  15  ;  but  had  A  travelled  five  hours  less,  and  B  had  gone  2 
miles  an  hour  more,  they  would  have  been  in  the  proportion 
of  2  :  5.  How  many  miles  did  each  go  per  hour,  and  how 
many  hours  did  they  travel  before  they  met  1 

Ans.  A  went  4,  and  B  6  miles  an  hour,  and  they  travelled 
10  hours  after  B  set  out. 

Prob.  18.  A  Farmer  hires  a  farm  For  245Z.  per  annum,  the 
arable  land  being  valued  at  2l.  an  acre,  and  the  pasture  at  28 
shillings  :  now  the  number  of  acres  of  arable  is  to  half  the 
excess  of  the  arable  above  the  pasture  as  28  :  9.  How  many 
acres  were  there  of  each  "?     • 

Ans.  98  acres  of  arable,  and  35  of  pasture. 

Prob.  19.  A  and  B  playing  at  backgammon,  A  bets  3s.  to 
2s.  on  every  game,  and  after  a  certain  number  of  games  found 
that  he  had  lost  17  shillings.  Now  had  A  won  3  more  from 
B,  the  number  he  would  then  have  won,  would  be  to  the  num- 
ber B  had  won,  as  5  to  4.     How  many  games  did  they  play  ? 

Ans.  9. 

Prob.  20.  Two  persons,  A  and  B,  can  perform  a  piece  of 
work  in  16  days.  They  work  together  for  4  days,  when  A 
being  called  off,  B  is  left  to  finish  it,  which  he  does  in  36  days 
more.     In  what  time  would  each  do  it  separately  ? 

Ans.  A  in  24  days,  and  B  in  48  days. 

Prob.  21.  Some  hours  after  a  courier  had  been  sent  from 
A  to  B,  which  are  147  miles  distant,  a  second  was  sent,  who 
wished  to  overtake  him  just  as  he  entered  B  ;  in  order  to 
which  he  found  he  must  perform  the  journey  in  28  hours  less 
than  the  first  did.  Now  the  time  in  which  the  first  travels  17 
miles  added  to  the  time  in  which  the  second  travels  56  miles, 
is  13  hours  and  40  minutes.  How  many  miles  does  each  go 
per  hour  ? 

Ans.  the  first  goes  3,  and  the  second  7  miles  an  hour. 

Prob.  22.  Two  loaded  wagons  were  weighed,  and  their 
weights  were  found  to  be  in  the  ratio  of  4  to  5.  Paris  of  their 
loads,  which  were  in  the  proportion  of  6  to  7,  being  taken  out, 


PRODUCING  SIMPLE  EQUATIONS.         173 

their  weights  were  then  found  to  be  in  the  ratio  of  2  to  3  ;  and 
the  sum  of  their  weights  was  then  ten  tons.  What  were  the 
weights  at  first  ?  Ans.  16,  and  20  tons. 

Prob.  23.  A  and  B  severally  cut  packs  of  cards  ;  so  as  to 
cut  off  less  than  they  left.  Now  the  number  of  cards  left  by 
A  added  to  the  number  cut  off  by  B,  make  50  ;  also  the  num- 
ber of  cards  left  by  both  exceed  the  number  cut  off,  by  64. 
How  many  did  each  cut  off?         Ans.  A  cut  off  11,  and  B  9. 

Prob.  24.  A  and  B  speculate  with  different  sums  ;  A  gains 
150/,  B  loses  50Z,  and  now  A's  stock  is  to  B's  as  3  to  2.  But 
had  A  lost  50/,  and  B  gained  lOOZ,  then  A's  stock  would  have 
been  to  B's  as  5  to  9.     What  was  the  stock  of  each  ? 

Ans.  A's  was  300/,  and  B's  350/. 

Prob.  25.  A  Vintner  bought  6  dozen  of  port  wine  and  3 
dozen  of  white,  for  12/.  12^.  ;  but  the  price  of  each  after- 
wards falling  a  shilling  per  bottle,  he  had  20  bottles  of  port, 
and  3  dozen  and  8  bottles  of  white  more,  for  the  same  sum. 
What  was  the  price  of  each  at  first  1      > 

Ans.  the  price  of  port  was  2s.  and  of  white  3s.  per  bottle. 

Prob.  26.  Find  two  numbers,  in  the  proportion  of  5  to  7, 
to  which  two  other  required  numbers  in  the  proportion  of  3  to 

5  being  respectively  added,  i\\e  sums  shall  be  in  the  propor- 
tion of  9  to  13  :  and  the  difference  of  those  sums  =16. 

Ans.  the  two  first  numbers  are  30  and  42  ;  the  two  others, 

6  and  10. 

Prob.  27.  A  Merchant  finds  that  if  he  mixes  sherry  and 
brandy  in  quantities  which  are  in  the  proportion  of^  to  1,  he 
can  sell  the  mixture  at  78s.  per  dozen  ;  but  if  the  proportion 
be  as  7  to  2,  he  must  sell  it  at  79  shiUings  a  dozen.  Required 
the  price  of  each  liquor. 

Ans.  the  price  of  sherry  was  81s.,  and  of  brandy  '72s.  per 
dozen. 

Prob.  28.- A  number  consisting  of  two  digits  when  divided 
by  4,  gives  a  certain  quotient  and  a  remainder  of  3  ;  when  di- 
vided by  9  gives  another  quotient  and  a  remainder  of  8.  Now 
the  value  of  the  digit  on  the  left-hand  is  equal  the  quotient 
which  was  got  when  the  number  was  divided  by  9  ;  and  the 
other  digit  is  equal  Jyth  of  the  quotient  got  when  the  number 
was  divided  by  4.     Required  the  number.  Ans.  71. 

•  Prob.  29.  To  find  three  numbers,  such,  that  the  first  with 
J  the  sum  of  the  second  and  third  shall  be  120  ;  the  second 
with  1th  the  difference  of  the  third  s^nd  first  shall  be  70  ;  and 
•|  the  sum  of  the  three  numbers  shall  be  95. 

Ans.  50,  65,  and  75. 

Prob.  30.  There  are  two  numbers,  such,  that  J  the  greater 
16* 


174  SOLUTION  OF  PROBLEMS,  &c. 

added  to  i  the  lesser  is  13  ;  and  if  ^  the  lesser  be  taken  from  J 
the  greater,  the  remainder  is  nothing.   What  are  the  numbers  ? 

Ans.  18,  and  12. 

Prob.  31.  There  is  a  certain  number,  to  the  sum  of  whose 
digits  if  you  add  7,  the  result  will  be  three  times  the  left-hand 
digit;  and  if  from  the  number  itself  you  subtract  18,  the  digits 
wiirbe  inverted.     What  is  the  number  ?  Ans.  53. 

Prob.  32.  A  person  has  two  horses,  and  a  saddle  worth 
10/ ;  if  the  saddle  be  put  on  the  Jirst  horse,  his  value  becomes 
double  that  of  the  second  ;  but  if  the  saddle  be  put  on  the  se- 
cond horse,  his  value  will  not  amount  to  that  of  the  Jirst  horse 
by  13/.     What  is  the  value  of  each  horse  ? 

Ans.  56  and  33. 

Prob.  33.  A  gentleman  being  asked  the  age  of  his  two  sons, 
answered,  that  if  to  the  sum  of  their  ages  18  be  added,  the  re 
suit  will  be   double  the   age  of  the  elder ;  but  if  6  be   taken 
from  the  difference  of  their  ages,  the  remainder  will  be  equal 
to  the  age  of  the  younger.     What  then  were  their  ages  ? 

Ans.  30  and  12. 

Prob.  34.  To  find  four  numbers,  such,  that  the  sum  of  the 
1st,  2d,  and  3d,  shall  be  13  ;  the  sum  of  the  1st,  2d,  and  4th, 
15  ;  the  sum  of  the  1st,  3d,  and  4th,  18  ;  and  lastly  the  sum 
of  the  2d,  3d,  and  4th,  20.  *  Ans.  2,  4,  7,  9. 

Prob.  35.  A  son  asked  his  father  how  old  he  was.  His 
father  answered  him  thus.  If  you  take  away  5  from  my 
years,  and  divide  the  remainder  by  8,  the  quotient  will  be  i 
of  your  age  ;  but  if  you  add  2  to  your  age,  and  multiply  the 
whole  by  3,  and  then  subtract  7  from  the  product,  you  will 
have  the  number  of  the  years  of  my  age.  What  was  the  age 
of  the  father  and  son  ?  Ans.  53,  and  18. 

Prob.  36.  Two  persons,  A  and  B,had  a  mind  to  purchase 
a  house  rated  at  1200  dollars  ;  says  A  to  B,  if  you  give  me  ^ 
of  your  money,  I  can  purchase  the  house  alone  ^  but  says  B 
to  A,  if  you  will  give  me  |th  of  yours,  I  shall  be  able  to  pur- 
chase the  house.     How  much  money  had  each  of  them  ? 

Ans.  A  had  800  and  B  600  dollars. 

Prob.  37.  There  is  a  cistern  into  which  water  is  admitted 
by  three  cocks,  two  of  which  are  exactly  of  the  same  dimen- 
sions. When  they  are  all  open,  five-twelfths  of  the  cistern  is 
filled  in  4  hours  ;  and  if  one  of  the  equal  cocks  be  stopped, 
seven-ninths  of  the  cistern  is  filled  in  10  hours  and  40  minutes. 
In  how  many  hours  would  each  cock  fill  the  cistern  ? 

Ans.  Each  of  the  equal  ones  in  32  hours,  and  the  other  in  24. 

Prob.  38.  Two  shepherds,  A  and  B,  are  intrusted  with  the 
charge  of  two  flocks  of  sheep.     A's  consisting  chiefly  of  ewes, 


INVOLUTION.  175 

many  of  which  produced  lambs,  is  at  the  end  of  the  year  in- 
creased by  80  ;  but  B  finds  his  stock  diminished  by  20  :  when 
their  numbers  are  in  the  proportion  of  8  :  3.  Now  had  A  lost 
20  of  his  sheep,  and  B  had  an  increase  of  90,  the  numbers 
would  have  been  in  the  proportion  of  7  to  10.  What  were 
the  numbers  ?  Ans.  A's  160,  and  B's  110. 

Prob.  39.  At  an  election  for  two  members  of  congress,  three 
men  offer  themselves  as  candidates  ;  the  number  of  voters  for 
the  two  successful  ones  are  in  the  ratio  of  9  to  8  ;  and  if  the 
first  had  had  7  more,  his  majority  over  the  second  would  have 
been  to  the  majority  of  the  second  over  the  third  as  12  :  7. 
Now  if  the  first  and  third  had  formed  a  coalition,  and  had  one 
more  voter,  they  would  each  have  succeeded  by  a  majority  of 
7.     How  many  voted  for  each  ? 

Ans.  369,  328,  and  300,  respectively. 


CHAPTER  VI. 

ON 

INVOLUTION  AND  EVOLUTION 

OF  NUMBERS,  AND  OF  ALGEBRAIC  aUANTITIES. 

213.  The  powers  of  any  quantity,  are  the  successive  products, 
arising  from  unity,  continually  multiplied  by  that  quantity.  Or, 
the  power  of  the  order  m  of  a  quantity,  m  being  a  whole  pos- 
itive number,  is  the  product  of  that  quantity  continually  mul- 
tiplied 772  —  1  times  into  itself,  or  till  the  number  of  factors 
amounts  to  the  number  of  units  in  that  given  power. 

214.  Involution  is  the  method  of  raising  any  quantity  to 
a  given  power  ;  Evolution,  or  the  extraction  of  roots,  being 
just  the  reverse  of  Involution,  is  the  method  of  determining  a 
quantity  which,  raised  to  a  proposed  power,  will  produce  a 
given  quantity. 

Note. — The  term  root  has  been  already  defintfi,  (Art.  12). 

§  I.  involution  of  algebraic  quantities. 

215.  It  has  been  observed,  (Art.  13),  that  the  powers  of  al- 
gebraic quantities  are  expressed  by  placing  the  index  or  expo- 
nent of  the  power  over  the  quantity. 


176 


INVOLUTION. 


Hence,  if  a  proposed  root  he  a  single  letter,  and  without  a  cO' 
eff,cient,  any  required  power  of  it  will  he  expressed  hy  the  same 
letter  with  the  index  of  the  power  written  over  it.  Thus,  the 
nth  power  of  a  is  z=a",  n  being  any  positive  number  whatever. 

216.  If  the  proposed  root  he  itself  a  povjer,  the  required  power 
will  he  obtained  hy  multiplying  the  index  of  the  given  power  into 
that  of  the  required  power.  Thus  the  mXh  power  of  a^,  or 
{a'')'"=a'"P  ;  for  since,  (Art.  213),  (a'')'"=:a^XaPXa'',   &c.  = 

^p+p+p+etc._^^m  ^ ^jj 

where  the  number  of  factors  a''  is  equal  to  m. 

217.  Also,  if  a  simple  quantity  he  composed  of  several  factors, 
it  can  he  raised  to  any  power  by  multiplying  the  index  of  every  fac- 
tor in  the  quantity  hy  the  exponent  of  the  power.  Thus  the  mth 
power  of  {a''h''c'),  or  (a''^>V)^  is  =:  a'""6'''V" ;  for  since  (Art. 
274),  (a^hfc')'"  =  (a'h'c')  X  (a^^'c''),  <fcc.  =  a^a^  .  .  .  &'*'  .  . 
cV  .  .  .  =  (aP)'"  X  [h'')'"  X  (c'-)'"  ;  .  .  .  .  (2  ) ; 
by  observing  that  in  each  of  these  products,  such  as  a^a^,  &c., 
or  h^h^,  &c.,  there  enter  7n  equal  factors. 

Cor.  Hence,  if  the  proposed  quantity  has  a  numerical  coeffi- 
cient, it  must  also  he  involved  to  the  required  power.  Thus  the 
fourth  power  of  Sa^"^  is  =3^a'^'^b^'^— 3x3x3  X3Xa%^= 
81  a^h^.  For  the  numerical  coefficient  is  in  this  case  the 
same  as  any  other  factor. 

ROOTS  AND  POWERS  OF  NUMBERS. 


1st. 

2d 

1 

1 

2 

4 

3 

9 

4 

16 

5 

25 

6 

36 

7 

49 

8 

64 

9 

81 

3d. 


4th. 


5th.     6th. 


1 
16 

81 
256 


1 

32 

243 

1024 

3125 


1 

64 
729 
4096 
15625 


7th. 


Square 
Root. 


1 

|128 

12187 

116384 

178125 


1.414213 
1.732 
2. 
125;625  '3125  15625  178125  2.236 
21611296  7776  46656  |279936  2.449 
343J2401 16807  117649,823543  2.646 
512  4096  327Q8  262144:2097152  2.828 
729  6561  59049  53144114782969  3. 


Cube 
Root. 


1 

1.26 

1.442 

1.587 

1.71 

1.817 

1.913 

2. 

2.08 


218.  Any  power  of  a  fraction  is  equal  to  the  same  power  of 
the  numerator  divided  hy  the  like  power  of  the  denominator. 

Thus  the  mth  power  o^  j,ox  \rA'^=—;  for  (t)"*=tX? 

a    .  /*  .   ,c/,v    aXaX«,  etc.     a"*         ,  , 

X  J,  &c.  =  (Art.  156),  j-^-^-^=-  :   where  the  imm- 


ber  of  factors  -  is  equal  to  m. 


INVOLUTION.  177 

.,.,.,  1  ,  /.    «^^^         /aPM\ 

And  mlike  manner  the  mtn  power  of  —^^  or  / — ~]mcz= 

c  c  \  c  c  / 

(c'»)'"((^'')'""~c""'f/'''"  *         ■  *  •  *  •  V  /• 

219.  Any  even  power  of  a  positive  or  negative  quantity ^  is  nc' 
cessarily  positive.  In  fact,  2m  being  the  formula  of  even  nxmv- 
bers,  we  have  (±a)2'«=[(±a)2]'»— (-|-a2jni__|_a2m   ^  ^  ^  (4^_ 

220.  Any  odd  power  of  a  quantity  will  have  the  same  sign  as 
the  quantity  itself.     For,  the  general  formula  of  odd  numbers, 

(Art.  Ill),  being  2'"  +  l,we  have  (±a)2»»+»=(±a)2«»x(±a) 
=  fl2mx  ±a=±a2m+i (5). 

The  involution  of  algebraic  quantities  is  generally  divided 
into  two  cases. 


CASE  I. 

To  involve  a  simple  algebraic  Quantity. 

RULE. 

221.  Raise  the  coefficient,  if  any,  to  the  required  power, 
then  muhiply  the  index  of  each  factor,  or  letter,  by  the  index 
of  the  required  power,  and  write  their  several  products  over 
their  respective  factors  ;  Let  the  quantities  thus  arising  be  an- 
nexed to  each  other  and  to  the  same  power  of  the  coefficient, 
prefixing  the  power  sign,  and  it  will  be  the  power  required. 
Or,  multiply  the  quantity  into  itself  as  many  times  less  one  as 
is  denoted  by  the  index  of  the  power,  and  the  last  product,  with 
the  proper  sign  prefixed,  will  be  the  answer. 

Ex.   1.   Required  the  square,  or  second  power  of  2ab. 

Here,  {^Zabf=zAXa^Xb'^^^a%'^.     Ans. 

Ex.  2.  What  is  the  cube  of  — 3a2Z>2? 

Here,  {-^a^h'^Y=  (Art.  220),  -C^aWf^-^l Xa^'^ Xb"^-^ 
z=~-27a%^.      Ans. 

Ex.  3.  What  is  the  4th  power  of  —2a^x^  ? 

Here,  (—2a3x^Y=  (Art.  219),  +{2a^cc^Y=l6xa^''^x^-'^= 
Ida^^x^.     Ans. 

Ex.  4.  What  is  the  cube,  or  third  power  of  abc  ?  < 

Here,  abc  XabcXabcz=  a  X  a  X  a  X  bxbxbXcXcXcz:^ 
a^b-^c^.     Ans. 

222.  When  the  quantity  to  be  involved  is  a  fraction,  raise  both 
the  numerator  and  denominator  to  the  power  proposed. 

Ex.  5.  Required  the  4th  power  of  — — . 

tia 


178  INVOLUTION. 


( 


b\_        6*    _     b*    _  ¥ 
2a)  """'~(2a)*~2*Xa*~16a*' 

Ex.  6.  What  is  the  4th  power  of  —7—?  Ans.  . 

^                    3a?  81a;* 

Ex.  7.  What  is  the  8th  power  of  2a^  ?  Ans.  256a^^. 

Ex.  8.  What  is  the  7th  power  of  —x  ?  Ans.  —x''. 

Ex.  9.  What  is  the  6th  power  of ?  Ans.  — . 

x^  x^^ 

C  -5 

Ex.  10.  What  is  the  5th  power  of  -?  Ans 


5  *  ■  3125* 

5a;  625a:* 

Ex.  11.  What  is  the  4tli  power  of  -—  ?  Ans. . 

^  7  2401 

Ex.  12.  Required  the  cube  of -z-  ?  Ans. —. 

So  27  0-^ 

Ex.  13.  Required  the  square  of  %a2^2?  Ans.  a*K 

Ex.  14.  Required  the  9th  power  of  —xyl  Ans.  — a;^^^. 

Ex.  15.  Required  the  0th  power  of  a;y  ?  Ans.  1. 
Ex.  16.  Required  the  4th  power  of  a-2  ? 

Ans,  a-  or  V 
a? 


CASE  II. 
To  involve  a  compound  algebraic  Quantity. 

RULE  I. 

223.  Multi-ply  the  given  quantity  continually  into  itself  as 
many  times  minus  one  as  is  denoted  by  the  index  of  the  power, 
as  in  the  multiplication  of  compound  algebraic  quantities  (Art. 
79),  and  the  last  product  will  be  the  power  required. 
Ex.   1.  What  is  the  square  of  a -J- 25? 
a+2b 
a-{-2b 

a^+2ah 
+2a6+4J2 


Square  =a2+4«6  4-462 


INVOLUTION.  179 

Ex.  2.  What  is  the  cube  of  a2— a2  ? 


a*— 2aV+a;* 

f£l  —  ^2 

Cube  =a6— 3a*a;24-3a2a;'t— a;6 


Ex.  3.  Required  the  fourth  power  of  a-{-3b. 

Ans.  a*+i2a^-{-54aH^-\-108aP-^8lhK 

Ex.  4.  Required  the  square  of  3a;2-|-2ir4-5. 

Ans.  9a:'t  +  12a;3  +  34a;2+20a;+25. 

Ex.  5.  Required  the  cube  of  3a;— 5. 

Ans.  27^3_i35^2_|_ 225a;— 125 

Ex.  6.  Required  the  cube  of  x^— 2a;+l. 

Ans.  x^—6x^-\-l5x*--20x^+l5x^—ex+l 

Ex.  7.  Required  the  fourth  power  of  2  + 3a;. 

Ans.  16  +  96a;+216a;^+216a3+81ar* 

Ex.  8.  Required  the  fifth  poiver  of  1  —2a;. 

Ans.  1  — 10>r+40a;2  — 80a;34-80a'*— 32a;*. 

Ex.  9.  Required  the  square  of  a+^+c-j-J. 

Ans.  a'2i-b^+c'^-^d^  +  2(ab  +  ac-\-ad+bc-^bd+cd). 

224.  In  the  invokition  of  a  binomial  or  residual  quantity  of 
the  form  a-^-b,  or  a — b  ;  the  several  terms  in  each  successive 
power  are  found  to  bear  a  certain  relation  to  each  other,  and 
observe  a  certain  law,  which  the  following  Table  is  intended 
to  explain. 


180 


INVOLUTION. 


TABLE  OF  THE  POWERS  OF  «  +  J. 


Powers. 

Mode  of  ex- 
pressing them. 

Powers  expanded. 

Square. 

{a  +  hf. 

a^-\-2ah-\-b'^. 

Cube. 

{a^hf. 

a3  +  3a25-}-3«52_|_^,3. 

4th  power. 

[a+hf. 

a^-\-^a?h-^6a?b'^-{-4.ah^-\-bK 

5th  power. 

(a-\-bf. 

a^-\-5a^b-{-\0aW+\0a^¥^5a¥ 
-\-b\ 

6th  power. 

(a  +  hf. 

^6  ^  6a5^  4- 15^4  ^>2_|_  20^3^3 
-\-\5a%^+Qab^-\-¥. 

The  successive  powers  of  a— 6  are  precisely  the  same  as 
those  of  a-\~b,  except  that  the  signs  of  the  terms  will  be  al- 
ternately +  and  — .  Thus,  the  ffth  power  of  a  —  b  is  a^— 
5fl!*5-|-10a3^,2„l0a;262-f-5a6*— Z/5. 

225.  In  reviewing  that  column  of  the  above  Table  which 
contains  the  powers  of  a+6  expanded,  we  may  observe, 

I.  That  in  each  case,  the  first  term  is  raised  to  the  given 
power,  and  the  last  term  is  b  raised  to  the  same  power ;  thus, 
in  the  square,  the  first  term  is  a^,  and  the  last  b"^ ;  in.  the  cube, 
the  first  term  is  a^,  and  the  last  b^  ;  and  so  on  of  the  rest. 

II.  That,  with  respect  to  the  intermediate  terms,  the  pow- 
ers of  a  decrease,  and  the  powers  of  b  increase,  by  unity  in 
each  successive  term.     Thus,  in  the  fifth  power,  we  have 

In  the  second  term, a'^b  \ 

third, a^b"^  ; 

fourth, a2^,3  . 

fifth, a¥  ', 

and  so  on  in  other  powers. 

III.  That  in  each  case,  the  coefficient  of  the  second  term  is 
the  same  with  the  index  of  the  given  power.  Thus,  in  the 
square,  it  is  2  ;  in  the  cube,  it  is  3  ;  in  the  fourth  power,  it  is 
4  ;  and  so  on  of  the  rest. 

IV.  That  if  the  coefficient  of  a  in  any  term  be  multiplied  by 
its  index,  and  the  product  divided  by  the  number  of  terms  to  that 
place,  this  quotient  will  give  the  coefficient  of  the  next  term. 
Thus,  in  the  fifth  power,  the   coefficient  of  a  in  the  second 


INVOLUTION.  181 


4  v>  5      oQ 
to  that  place  =— - — =—-=10=  coefficient  of  the  third  term. 

2  3 


term  multiplied  by  its  index,  and  divided  by  the  number  of  terms 
—=10=  coefficient  of  the  third  ten 

i 
Tv,  *!,«  „;^+l,  «^..r««  CoefF.  of  a  in  the  4th  term .  its  index      20  X  3 

In  the  sixth  power,         r — tt r— ti.— :— i =  — : — = 

^  number  of  terms  to  that  place.  4 

fin 

—-=15=  coefficient  of  the  fifth  term. 

Hence,  we  are  furnished  with  the  following  general  ruler  for 
raising  a  binomial  or  residual  quantity  to  any  power,  without 
the  process  of  actual  multiplication.  •  * 

RULE  II. 

226.  Find  the  terms  without  the  coefficients,  by  observing 
that  the  index  of  the  first,  or  leading  quantity,  begins  with  that 
of  the  given  power,  and  decreases  continually  by  1,  in  every 
term  to  the  last ;  and  that,  in  the  following  quantity,  its  indices 
are  1,  2,  3,  &c.  Then,  find  the  coefficients,  by  observing  that 
those  of  the  first  and  last  terms  are  always  1  ;  and  that  the 
coefficient  of  the  second  term  is  the  index  of  the  power  of  the 
first ;  and,  for  the  rest,  if  the  coefficient  of  any  term  be  mul- 
tiplied by  the  index  of  the  leading  quantity  in  it,  and  the  pro- 
duct be  divided  by  the  number  of  terms  to  that  place;  it  will 
give  the  coefficient  of  the  term  next  following. 

Ex.  1.  Required  the  8th  power  oi  a-^b. 
Here  the  terms,  without  the  coefficients,  are 

«8,  a^b,  a%-\  a^b\  a^b\  a?b\  a?b\  ab\  b^. 
And  the    coefficients,  according  to  the  rule,  will  be.  1,  8, 

^=28,?5><l=56,^=70,I52<i=56,^  =  28. 
2  3  4  5  o 

28X2     ^    8X1      , 

Then,  the  terms  are  thus  : 

The  first  term  is c?. 

second, ScPb. 

third,        ....       ?^Xa662=:28a«62. 
2 

fourth,      ....     ^?^Xa5&3^56a553. 

fifths         ....     ^^^xa^b^  =  10a^b\.. 

sixth,        ....     ^-^^Xa^b^=5Qa%K 
5 
17 


182  INVOLUTION.    . 

56X3 
seventh,    ....     — - — Xa%^=2Ba'^h^ 
o 

2S  y  3 
eighthy      ....  XalP  =  8a  J^ 

ninth,        ....       ?4^X     &«=       58. 

o 

And  thus  we  have,  (0+5)8=084. 8a'^i+ 280^^2^ 56aSJ3^ 
70a*5*+56a355^_28a256+8a574-58. 

227.  From  this  example  and  the  foregoing  Table  the  whole 
number  of  terms  will  evidently  be  one  more  than  the  index  of 
the  given  power ;  after  having  calculated  therefore  as  many 
terms  as  there  are  units  in  the  index,  of  the  given  power,  we 
may  immediately  proceed  to  the  last  term.  And  in  like  man- 
ner it  may  be  observed,  that  when  the  number  of  terms  in  the 
resulting  quantity  is  even,  the  coefficients  of  the  two  middle 
terms  is  the  same  ;  and  that  in  all  cases  the  coefficients  in- 
crease as  far  as  the  middle  term,  and  then  decrease  precisely  in 
the  same  manner  until  we  come  to  the  last  term.  By  attend- 
ing to  this  law  of  the  coefficients,  it  will  be  necessary  to  cal- 
culate them  only  as  far  as  the  middle  term,  and  then  set  down 
ihe  rest  in  an  inverted  order. 

Thus  in  the  above  example,  the  middle  term  is  70a*6*,  and 
we  have, 

T\ie  first  four  coefficients,     1,     8,  28,  56. 
The  last  four     ....  56,  28,    8,     1. 

228.  But  we  are  not  yet  arrived  at  the  most  general  form  in 
which  this  Rule  may  be  exhibited.  Suppose  it  was  required 
to  raise  the  binomial  a+5  to  any  power  denoted  by  the  num- 
ber (n).  Proceeding  with  n  as  we  have  done  with  the  several 
indices  in  the  preceding  examples,  it  appears  that, 

The  first  term  would  be  a". 
The  second,      .     .     .     na^—^h. 

The  third,        .     .     .     -^ — '-an-ib. 

rxM.     r      .L                         n(n—l)x(n—2)    ,^,,, 
The  fourth,      .     .     .     -i ^^ ^o»--3R 

The  «>M.  "^"-^^"^rxsxir/'^'""'-"-"^- 


The  last, b\ 


INVOLUTION.  •  183 

Or  (<z4-^>V=g"+ng"-'54-''^'^~^W&^+ 

n{n-l)x{n-2)  n(n-l)x(n-2)x(n-3) 

2.3  "•"  2.3.4 

ar-^b\SLc +&". 

By  the  same  process,         {u—bY=a'' — 7ia''~'^-i- 

n(7i  — 1)  ^  „,     n{n—l)x{n—2)   ^.,.   , 
-J ^z"~2J2 J^ i — i i  a»-3p  ^ 

4)  4i.O 

•"^ o  o  o        iflr~*5* — &c. ;  the  signs  of  the  terms 

.^.0.-3 

being  alternately  +  and  —1  ;  and  the  sign  of  the  last  term  is 

+  or  —  1 ,  according  as  n  is  even  or  odd  ;    we  have  the  last 

term  in  the  former  case^  -\-h\  and  in  the  latter  —  &". 

This  general  and  compendious  method  of  raising  a  binomial 
quantity  to  any  given  power,  is  called  from  the  name  of  its  ce- 
lebrated inventor,  Sir  Isaac  Newton's  "  Binomial  Theorem." 
The  demonstration  of  this  Theorem,  with  its  application  to  the 
finding  the  powers  and  roots  of  compound  quantities,  forms 
the  subject  of  another  Chapter.  Its  present  use  will  appear 
from  the  following  Example. 

Ex.  2.   Required  the  fifth  power  of  x'^-\''^y^. 

Substituting  these  quantities  for  *a,  5,  w,  in  the  foregoing 
general  formula,  it  appears  that 

''):£' V'^)    ■  ■  ■  ■  M'  •  -  -  •  •  -"• 

2nd,.     .    {naJ^-^b)    .     is  5  X  (a:^)^  x  3y2      .     .     ,    =15a:y. 
3d,     .      ^^(^~^)^n_2^2\    .    5x|x(^2)3x(3/)2=90a;y. 

4th,    .      (±^±Z-^^-.b.)   ,  is5x|x|x<^)^X 

(3y2)3 =27Wy\ 

/n(w— l)(n— 2)(n  — 3)      ,,  \  .    ^432 
5th,    .       (J \-^ -V-3^)is5x-X-X-x^^X 

(3y2J4 =405^y. 

Last,  .  (6")  is  (3y2)5  ,  ^243y»o. 

So  that  (a:2  +  3y2)5  =  jpio  ^  I5a;8y2  _j_  80a;y  +  270a?*/  4- 
405a;y  +  243yi'^. 

229.  By  means  of  this  Theorem,  we  are  enabled  to  raise  a 
trinomial,  or  quadrinomial  quantity  to  any  power,  without  the 
process  of  actual  multiplication. 

Ex.3.  Required  the  square  of  a+^  +  <'- 

Here,  including  a-\-b  in  a  parentheses  {a-\-h),  and  consider- 
ing it  as  one  quantity,  we  should  have  {a-\-h-{-cf-=z\{a-\'b\ 
-f  cp  ;  and  comparing  them  with  the  general  formula  ; 


184  •  EVOLUTION. 

we  have  (a")  =  (a+bY=za'^+2ab-{-b^  ) 
(na''-^Y=2{a-{-l))xc=2ac-{-2bc  l 
{b")=:c^  -  =c2  ) 

Hence,  {a+b  +  cf=(a+bf-\-2(a-]-b)Xc-^c^=a^+2ab  + 

Ex.  4.  Required  the  seventh  power  of  a — b. 

Ans.   a^—7a^^+  2la^b'^—35a^b^  +  35a^*—2la'^b^  +  7ab^ 

Ex.  5.  Required  the  sixth  power  of  3a;  +  2y. 
Ans.    729a;6+2916a:5y4-4860a:y+4320a;y4-2160a;y-f 
576xy^-{-e4f. 

Ex.  6.  Required  the  square  oi  x-\-t/-\-3z. 

Ans.  oe^-{-2ocy-^y'^-^6xz-\-6yz-\-9z\ 
Ex.  7.  Required  the  fifth  power  of  14-2«. 

Ans.  l-i-10a;+40a^+80ar3  +  80ir*+32ar^. 
Ex.  8.  Required  the  cube  of  x^—2xy-{-y^. 

Ans.  x^—6x^7/i-l5x*f—20x^y^-^15x'^y*-'6xi/^+y^. 

^  II.     EVOLUTION  OF  ALGEBRAIC  QUANTITIES. 

230.  The  quantity  which  has  been  raised  to  any  power  is  call- 
ed the  root  of  that  power ;  thus  the  with  root  of  a  power,  is  that 
quantity  which  we  must  continually  multiply  into  itself,  till 
the  number  of  factors  be  equal  to  m,  m  being  a  positive  whole 
number,  in  order  to  produce  the  power  proposed.  We  may 
conclude  from  this  definition,  and  from  the  Articles  in  the  pre- 
ceding section. 

231.  That  the  mih  root  of  a  quantity  such  as  a^,pm  being  a 
multiple  of  p,  is  obtained  by  dividing  the  exponent  pm  of  this 
quantity,  by  the  index  of  the  required  root.    Thus  the  mth  root  of 

pm  g 

a^=a'"  =a^  ;  the  square  root  of  a^=a^  =  a?,  and  the  cube 

6. 

ropt.of  a^=a^=a^. 

232.  Also  that  the  mih.  root  of  a  product  such  as  ai^b^,  is 
equal  to  the  mih.  root  of  each  of  its  factors  multiplied  together. 
Thus,  the  mth  root  of  a^P'"  is  =  the  mth  root  of  a^  x  the  mth 

root  of  P'"=a~^  X  b'^=  a^^. 

233.  And  that  the  mth  root  of  a  fraction  such  as  -r-,  is  equal 

to  the  mih  root  of  the  numerator  divided  by  the  mih  root  of  its 
denominator. 

Thus  the  mth  root  of  t-  = — =  -. 
b"*       "L     b 


EVOLUTION.  185 

234.  The  square^  the  fourth  root,  or  any  even  root  of  an  affir- 
mative quantity  may  he  either  •{■  or  —  1 .  Thus  the  square  root 
of  a^  —  ^^r  _fl5  J  for  -|-ax+«=^-a^  and  —ay.—a=i-\-(jP-. 
In  fact,  the  2 with  root  of  cP-^l^  equal  to  +  a  or  —o  J  for  {^-^af^ 

235.  Any  odd  root  of  a  quantity,  will  have  the  same  sign  as  the 
quantity  itself.  Thus  the  (2m4-  l)th  root  of  ia^*"  +^  is  equal  to 
±a;  for  (±a)2'"+i  is  equal  to  ia^^+i. 

236.  Evolution,  or  the  rule  for  extracting  the  root  of  any 
algebraic  Quantity  whatever,  is  divided  into  the  four  following 
Cases. 


CASE  I. 
To  find  any  root  of  a  simple  algebraic  Quantity. 

RULE. 

237.  Extract  the  root  of  the  coefficient  for  the  numeral  part, 
and  the  root  of  the  quantity  subjoined  to  it  for  the  literal  part, 
by  the  methods-  pointed  out  in  the  above  propositions  ;  then, 
these,  joined  together,  will  be  the  root  required. 

Ex.  1.    It  is  required  to  find  tl;e  square  root  of  x^. 

4 

Here,  the  square  root  of  a:^=  J;-\/a;*z:r  J-a:2  =  _[-a;2. 
Ex.  2.  Required  the  cube  root  of  — 21x^a^. 
Here,  the  cube  root  of  —21x^a^=—^  21x^a^=:  — ^  27  X 
^  a;3  X  3/ a6_  _3  X  a;  X  a2=  _3a2a;. 

Ex.  3.  Required  the  square  root  of  -r--. 

o^c^ 

Here,  the  square  root  of  a'^x^=:\/a'^x-\/x^:=ax,  and  the 

square  root  of  ¥'c'^=i^h'^x  \/c^=bc  ;  .*.  i^—  is  the  root  re- 

bc 

quired. 

Ex.  5.  It  is  required  to  find  the  square  root  of  Sia^x*. 

Ans.  &ax^,  or  —8ax\ 

Ex.  6.  It  is  required  to  find  the  cube  root  of  729a^x^^. 

Ans.  90%*. 

Ex.  7.  Required  the  fourth  root  of  25*6a*5^. 

Ans.  4ah^,  or  —4ab^. 

Ex.  8.  Required  the  fifth  root  of  32a^x^^.  Ans.  2ax\ 

Ex.  9.  Required  the  sixth  root  of  tttttx-t^.        Ans.  4- — -. 

4096a;^2  4a:* 

.    17* 


186  EVOLUTION. 

Ex    10.  Required  the  ninth  root  of  — rf^.      Ans. v. 

a^o^  ah 

Ex.  11.  -Required  the  square  root  of -— .  Ans.  4-- . 

*  ^  4a;2y2  =»=  2xy 

o4^  4/17 

Ex.  12.  Required  the  cube  root  of  r—-rrT'  Ans.  —-^. 

CASE  11. 
To  extract  the  square  root  of  a  compound  .Quantity. 

RULE 

238.  Observe  in  what  manner  the  terms  of  the  root  may  be 
derived  from  those  of  the  power  ;  and  arrange  the  terms  ac- 
cordingly ;  then  set  the  root  of  the  first  term  in  the  quotient ; 
subtract  the  square  of  the  root,  thus  found,  from  the  first 
term,  and  bring  down  the  next  two  terms  to  the  remainder  for 
a  dividend. 

Divide  the  dividend,  thus  found,  by  double  that  part  of  the 
root  already  determined,  and  set  down  the  result  both  in  the 
quotient  and  divisor. 

Multiply  the  divisor,  so  increased,  by  the  term  of  the  root 
last  placed  in  the  quotient,  and  subtract  the  product  from  the 
dividend,  and  to  the  remainder  bring  down  as  many  terms  as 
are  necessary  for  a  dividend,  and  continue  the  operation  as  be- 
fore. 

Ex.  1.  Required  the  square  root  of  a'^-\-2ab-^b^, 
a^+2'ab-\-b^ 


2a  4-^ 


2aJ+62 
2aJ+52 


On  comparing  a-\-b  with  c^-\-2ab-{-b'^,  we  observe  that  the 
first  term  of  the  power  (a-)  is  the  square  of  the  first  term  of 
the  root  (a).  Put  a  therefore  for  the  first  term  of  the  root, 
square  it,  and  subtract  that  square  from  the  first  term  of  the 
power.  Bring  down  the  other  two  terms  2ab'\-b'^^  and  double 
the  first  term  (a)  of , the  root ;  set  down  2a,  and  having  divi- 
ded the  first  term  of  the  remainder  (2ai)  by  it,  we  have  5,  the 
other  term  of  the  root ;  and  since  2ab-\-b'^  —  {^a-\-h)  xb,  if  to 
2a  the  term  b  is  added,  and  this  sum  multiplied  by  5,  the  re- 
sult is  2ab-\-b'^ ;  which  being  subtracted  from  the  terms  brought 
down,  nothing  remains. 


EVOLUTION.  187 

Ex.  2.  Required  the  square  root  of  a2+2a5+52+2ac4-2dc 

a^-{-2ab+b^-\-2ac+2bc-{-c\a-{-b+c 
a2 


2a+i 


2ab+b^ 
2ab+b^ 


2a-\-2b-\-c 


2ac+25c4-c2 
2ac4-2ic+c2 


On  comparing  the  root  a-j-^+c,  thus  found  with  its  power, 
the  reason  of  the  rule  for  deriving  the  root  from  the  power 
is  evident.  And  the  method  of  operation  is  the  same  as  in  the 
last  example.  Thus,  having  found  the  first  two  terms  of  the 
root  as  before,  we  bring  down  the  remaining  three  terms  2ac 
4-26c+c2  of  the  power,  and  dividing  2ac  by  2a,  it  gives  c,  the 
third  term  of  the  root.  Next,  let  the  last  term  {b)  of  the  pre- 
ceding divisor  be  doubled,  and  "add  c  to  the  divisor  thus  in- 
creased, and  it  becomes  2a-{-2b-\-c  \  multiply  this  new  divisor 
by  c,  and  it  gives  2ac-\-2bc-{-c'^,  which  being  subtracted  from 
the  terms  last  brought  down,  leaves  no  remainder.  In  like 
manner  the  following  Examples  are  solved. 

89 
Ex.  3.  Required  the  square  root  o{  Ax^-\-&x^-{-~x'^-{-  Ibx 

-^25. 

89  /  3 

4«4-f6aj3-f— a;2+15a;-f-25^2a;2-f--a:+5 

4x* 


4 


2  /  4 


6x^-\--x^ 
4 


4a:2+3a;+5)20.T24-l5a;+25 
20a:2-f  15a;+25 


Ex.  4.  Required  the  square  root  of  a;^-|- 4a^-}- 2a;* +9i»^— 4a; 
+4.  Ans.  a:3+2a;2— a;+2. 

Ex.  5.  Required  the  square  root  of  x'^-\'Aax^+&a'^x^'\-^a^x 
-f-a*.  Ans.  a;2-}-2aa;-|-a2. 

Ex.  6.  Required  the  square  root  oia'*'—2a^-{-^a'^—\a+  ^. 

•  Ans.  a^ — «+^' 


188  EVOLUTION. 

Ex.  7.  Required  the  square  root  of  Aa^+l2a^x-\-l3a'^x^-\' 
eax^-^xK  Ans.  2a'^-\-3ax-{-x'^. 

Ex.  8.  Required  the  square  root  of  9a?* +12a;3+34a;2+ 20a: 
+25.  Ans.  3a?2^2a:+5. 

Ex.  9.  Required  the  square  Voot  of  a^-\-2ab+b^-\-2ac-\- 
'  2ic+c2+2a(f4  2bd-\-2cd+d^.  Ans.  a-]-b+c+d. 

Ex.  10.  Required  the  square  root  of  a*+12a3^-f54a262+ 
I08ab^-^8lb*.  Ans.  a^-\-6ab  +  9b\ 

Ex.  11.  Required  the  square  root  of  a^  —  da^x+lda'^x^— 
20a^x^+l5a^x'^—6ax^-\-jic^.  .    Ans.  a^—Sa^x+Sax^—x^. 

Ex.  12.  Required  the  square  root  of  a'^—2a'^x^-\-x*. 

Ans.  a^—x^ 

CAS-E  III. 

To  extract  the  cube  root  of  a  compound  Quantity, 

RtJLE. 

239.  Arrange  the  terms  as  in  the  last  case  ;  and  set  the 
root  of  the  first  terms  in  the  quotient ;  subtract  the  cube  of  the 
root,  thus  found,  from  the  first  term,  and  bring  down  three 
terms  for  a  dividend. 

Next,  divide  the  first  term  of  the  dividend  by  3  times  the 
square  of  that  part  of  the  root  already  determined,  and  set  the 
result  in  the  quotient ;  then,  to  .3  times  the  square  of  that  part 
of  the  root,  annex  3  times  the  product  of  the  same  part  and  the 
last  result,  and  also  the  square  of  the  last  result,  with  their  pro- 
per signs  ;  and  it  will  give  the  divisor,  multiply  the  divisor  by 
the  term  of  the  root  last  placed  in  the  quotient,  and  subtract 
the  product  from  the  dividend,  bring  down  three  terms  or  as 
many  as  may  be  necessary  for  a  dividend,  and  proceed  as  be- 
fore. 

Ex.  1.  Required  the  cube  root  of  a3  4- 3a2J+ 30^24.^3. 

a^-i-Sa^+Sab^-hP 

a3  (a+& 


3a^-\-3ab-^b^)3a^b+3ab^+b^ 
3a25-f3a62_j_J3 


The  reason  of  the  rule  may  be  made  evident  from  a  com- 
parison of  the  roots  with  its  cube. 

Or,  thus,  if  the  quantity  whose  root  is  to  be  extracted,  has 
an  exact  root,  the  root  of*  the  leading  term  must  be  one  term 


EVOLUTION.  189 

of  its  root ;  that  is,  the  cube  root  of  a^,  which  is  a,  is  one 
term  of  the  root,  and  the  remaining  terms  being  brought 
down,  the  root  of  the  last  term  P  is  consequently  another  term 
of  the  root ;  but  as  the  root  may  consist  of  more  terms  than 
two  ;  the  next  term  (b)  of  the  root  is  always  found  by  dividing 

(——=b\  the  first  term  of  the  dividend  by  three  times  the 

square  of  the  divisor,  and  the  two  remaining  terms  of  the  di- 
vidend 3aP-\-b3={3ab-{-b^)b  ;  hence  3ab-{-b^  must  be  added 
to  3a^  for  a  divisor  ;  and  so  on. 

Ex.  2.  Required  the  cube  root  of  x^-[-6x^—40x^-\-  96a;— 64. 

a;6_|_6^5_40a;3-f  96a;— 64  (a;2+2a;— 4 

a;2 


3a;*+ 6a;3  +  4a;2)6a;'^ — 40a;3 

6a;5+12a;*+8a;3 


3a;*+12.'r3— 24a;+ 16)  — 12a;4— 48x3+960;— 64 
—  12a;*— 48a;3  +  96a;— 64  ^ 


Ex.  3.  Required  the  cube  root  of  (a'\-bY-{-3{a-\-bYc+ 
3(a+&)c2-fc3.  Ans.  a+6+c. 

Ex.  4.  Required  the  cube  root  of  a;^— 6a;^+15a;*— 20a;3-|- 
15a;2— 6a;+l..  Ans.  a;2— 2a;+l. 

Ex.  5.  Required  the  cube  root  o(  x^-\-6x^i/-\-l5oc^y^-{-20x^^^ 
+  15acVH-6a;yHy6.  Ans.  a;24-2a;y+y2, 

Ex.  6.  Required  the  cube  root  of  1— 6a;+12a;2— Sa;^. 

Ans.  1— 2a;. 

CASE  IV. 
To  find  any  root  of  a  compound  Quantity. 

RULE. 

240.  Find  the  root  of  the  first  term,  which  place  in  the  quo- 
tient ;  and  having  subtracted  its  corresponding  power  from 
that  term,  bring  down  the  second  term  for  a  dividend.  Divide 
this  by  twice  the  part  of  the  root  above  determined,  for  the 
square  root ;  by  three  times  the  square  of  it,  for  the  cube  root ; 
by  four  times  the  cube  of  it,  for  the  fourth  root,  &c.  and  the 
quotient  will  be  the  next  term  of  the  root. 

Involve  the  whole  of  the  root,  thus  found,  to  its  proper 
power,  whi^  subtract  from  the  given  quantity,  and  divide 
the  first  term  of  the  remainder  by  the  same  divisor  as  before. 


190  EVOLUTION. 

Proceed  in  the  same  manner  for  the  next  following  term  of  the 
root ;  and  so  on,  till  the  whole  is  finished. 

241.  This  rule  may  be  demonstrated  thus;  {a-{-bf=(f 
•^na"-^b+,  &c.  Here  the  nth  root  of  a"  is  a,  and  the  next 
term  na"-^b  contains  b,  (the  other  term  of  the  root)  na"-'^ 
times  ;  hence,  if  we  divide  na"-'^b  by  na"-^,  we  have  5,  or 


na 


;j-^  =b  ;  and  so  on,  for  any  compound  quantity,  the  root 

of  which  consists  of  more  than  two  terms. 

Now,  if  n=:2  ;  then,  the  divisor  na'*-^=2a,  for  the  square 
root; 

if  71=3;  then,     .     ,     .     .     na"-^=3a^,  for  the  cube 
root ; 

ifn=4;  then,     ....     na"-^=4a^,   for  the   4th 
root; 

ifw=5;  then,     ....     na"--^z=5a*,  for  the   5th 
root. 

And  so  on  for  any  other  root,  that  is,  involve  the  first  term 
of  the  root,  to  the  next  lowest  power,  and  multiply  it  by  the 
index  of  the  given  power  for  a  divisor. 

Ex.  1.  Required  the  square  rootof  a*— 2a^a:-f'3a^a;^ — 2ax^ 
+  a:*. 


2a'^)—2a^x 


(a'^—ax)^=a^—2a^x-\-a'^x^ 


2a2)-f.2a2a;2 


{a^  —  aX'\-x^Y=a^~2a^x+3a'^x'^—2ax^+x^. 


Ex.  2.    Required  the  4th  root  of  l6a*  —  96a^x-\-2l6a^x^— 
2ieax^-{-Slx\ 

l6a\—96a3x-\-2l6a2x^—2l6ax^  +  8lx^(2a—3x 
16a4 


4x{2ay  =  32a^)  —  96a^x 


(2a'-3xY  =  l6a*—9ea^x-\-2l6a^x^^2ieax^-{-8lx^. 

242.  As  this  rule,  in  high  powers,  is  often  found  to  be  very 
laborious,  it  may  be  proper  to  observe,  that  the  woots  of  cer- 
tain compound  quantities  may  sometimes  be  easily  discovered ; 


EVOLUTION.  191 

thus,  in  the  last  example,  the  root  is  2a— 3a?,  which  is  the 
difference  of  the  roots  of  the  first  and  last  terms ;  and  so  on, 
for  other  compound  quantities. 

Hence,  the  following  method  in  such  cases  ;  extract  the 
roots  of  all  the  simple  terms,  and  connect  them  together  by 
the  signs  +  ^^  — ,  as  may  be  judged  most  suitable  for  the 
purpose ;  then  involve  the  compound  root  thus  found,  to  its 
proper  power,  and  if  it  be  the  same  with  the  given  quantity, 
it  is  the  root  required.  But  if  it  be  found  to  differ  only  in 
some  of  the  signs,  change  them  from  +  to  — ,  or  from  —  to 
+  ,  till  its  power  agrees  with  the  given  one  throughout.  How- 
ever, such  artifices  are  %ot  to  be  used  by  learners,  because 
the  regular  mode  of  proceeding  is  more  advantageous  to  them  ; 
besides,  a  knowledge  of  those  artifices  which  are  used  by  ex- 
perienced Algebraists,  can  only  be  acquired  from  frequent 
practice. 

Ex.  3.  Required  the  square  root  of  a'^-\-2ah-\-h'^-\-2ac-\-2hc 

Here,  the  square  root  of  a'^=za  ;  the  square  root  of  Ir^zzih  ; 
and  the  square  root  of  c'^^:zc.  Hence,  a-\-b-\-c,  is  the  root  re- 
quired, because  {a-\-b-{-cY=ia'^-\-2ah-\-b'^-\-2ac-{-2bc-fc'^. 

Ex.  4.  Required  the  fifth  root  of  32a;5— SOar^+SOx^  — 40a;2 
+  10a:— 1.  Ans.  2a;— 1. 

Ex.  5.  Required  the  cube  root  oi  x^—Qx^-\-\bx^'—20x^-{^ 
\bx^--bx^\-\.  *  Ans.  ar^— 2a;-l-l. 

Ex.  6.  Required  the  fourth  root  oi  a^~A.a^x-\-Qa'^x'^-^Aa!x^ 
4-a:*.  Ans.  a—x 

Ex.  7.  Required  the  square  root  o{  x^-{-2x^y^-\-y^. 

Ans.  a:*4"y* 

Ex.  8.  Required  the  square  root  of  a;^— 2a;y-f-y^. 

Ans.  x^—y^. 

Ex.  9.  Required  the  cube  root  of  a^-. Ga^a? 4-12 aa;2—8a;3. 

Ans.  a—2x, 

Ex.  10.  Required  the  sixth  root  of  a;^— 6a;S4-15a;'*— 20a;3+ 
15a:2  — 6a?+l.  Ans.  a;  — 1. 

Ex.  11.  Required  the  fifth TOOt  of  x^^-\'\bx^y'^-\-9Qx^y^-\- 
270a:y+405a:y+243yio.  Ans.  a;2  +  3y2. 

Ex.  12.  Required  the  square  root  of  x^-^2xy-\-y'^-\-Qxz-\' 
Gyz-{-92^.  Ans.  x-{-y+3z. 

^   HI.  INVESTIGATION  OF  THE  RULES  FOR  THE  EXTRACTION 
OF  THE  SQUARE  AND  CUBE  ROOTS  OF  NUMBERS. 

243.  It  has  been  observed,  (Art.  104),  that,  a  denoting  the 
tens  of  a  number,  and  b  the  units,  the  formula  a'^-\-2ab-\-b^ 
would  represent  the  square  of  any  number  consisting  of  two 


192  EVOLUTION. 

figures  or  digits  ;  thus,  for  example,  if  we  had  to  square  25 

put  a=20  and  5=5,  and  we  shall  find 

^2^400 

2ab=200^ 

b^=  25 


{a+bf  =  (25f=:625. 

244.  Before  we  proceed  to  the  investigation  of  these  Rules, 
it  will  be  necessary  to  explain  the  nature  of  the  common 
arithmetical  notation.  It  is  very  well  known  that  the  value 
of  the  figures  in  the  common  arithmetical  scale  increases  in  a 
tenfold  proportion  from  the  right  to  ftie  left ;  a  number,  there- 
fore, may  be  expressed  by  the  addition  of  the  units,  tens,  hun- 
dreds, &LC.  of  which  it  consists  ;  thus  the  number  4371  may  be 
expressed  in^the  following  manner,  viz.  40004-300-|-70-|-l, 
or  by  4x  10004-3 X  100+7X  10-j-l  ;  also,  in  decimal  arith- 
metic, each  figure  is  supposed  to  be  multiplied  by  that  power 
of  10,  positive  or  negative,  which  is  expressed  by  its  distance 
from  the  figure  before  the  point :  thus,  672..53  =  6  X  102+7  X 
10^+2x100+5x10-1  +  3x10-2  =  6x100  +  7x10+2x1 

+  —  +  —-=672+— -+-^-  =  672-^1.     Hence,  if  the  digits 
^10^100  ^100     100  100  '  ^ 

of  a  number  be  represented  by  a,  h,  c,  d,  e,  &c.  beginning 

from  the  left-hand  ;  then, 

A  number  of  2  figures  may  be  expressed  by  \Oa-\-h. 

3  figures     ...      by  lOOa+106+c. 

4  figures     .  by  1000a+100Z>+10c+d 
&c.  &c.  &c. 

By  the  digits  of  a  number  are  meant  the  figures  which  com- 
pose it,  considered  independently  of  the  value  which  they 
possess  in  the  arithmetical  scale. 

Thus  the  digits  of  the  number  537  are  simply  the  numbers 
5,  3  and  7  ;  whereas  the  5,  considered  with  respect  to  its 
place,  in  the  numeration  scale,  means  500,  and  the  3  means  30. 

245.  Let  a  number  of  three  figures,  (viz.  lOOa+105  +  c) 
be  squared,  and  its  root  extracted  according  to  the  rule  in  (Art. 
288),  and  the  operation  stands  thus  ; 

I.     10000a2  +  2000a&+  \0Qb'^^200ac-\-20hc-\-c'^ 
10000^2  (lOOa+106+c 


200a+ 1 0b)2Q00ab  + 1 005^ 
2000a5+100& 


200a+20J+c)200ac+205c+c2 
200ac+206c+c2 


EVOLUTION.  193 


*     ^   ^— s  V    ^^^  ^^®  operation  is  transformed  into  the 


_  following  one  , 

40000+12000+9004.400+60+1(200+30+1 
40000 


400+30)12000+900 


400+60+1)400+60+1 
400+60+1 


III.  But  it  is  evident  that  this  operation  would  not  be  af- 
fected by  collecting  the  several  numbers  which  stand  in  the 
same  line  into  one  sum,  and  leaving  out  the  ciphers  which  are 
to  be  subtracted  in  the  operation. 

53361(231 
4 


43 

461 


133 

129 

461 
461 


Let  this  be  done  ;  and  let  two  figures  be  brought  down  at  a 
time,  after  the  square  of  the  first  figure  in  the  root  has  been  sub- 
tracted ;  then  the  operation  may  be  exhibited  in  the  manner 
annexed  ;  from  which  it  appears,  that  the  square  root  of  53361 
is  231. 

246.  To  explain  the  division  of  the  given  number  into  pe- 
riods  consisting  of  two  figures  each,  by  placing  a  dot  over 
every  second  figure  beginning  with  the  units,  as  exhibited  in 
the  foregoing  operation.  It  must  be  observed,  that,  since  the 
square  root  of  100  is  10  ;  of  10000  is  100  ;  of  1000000  is  1000 ; 
(fee.  &c.  it  follows,  that  the  square  root  of  a  number  less  than 
100  must  consist  of  one  figure  ;  of  a  number  between  100  and 
10000,  of  two  figures  ;  of  a  number  between  10000  and 
1000000,  oi  three  figures  ;  &c.  &c.,  and  consequently  the  num- 
ber of  these  dots  will  show  the  number  of  figures  contained 
in  the  square  root  of  the  given  number.  From  hence  it  fol- 
lows, that  \hefi.rst  figure  of  the  root  will  be  the  greatest  square 
root  contained  in  the  first  of  those  periods  reckoning  from  the 
left. 

Thus,  in  the  case  of  53361  (whose  square  root  is  a  num- 
18 


194  EVOLUTION. 

ber  consisting  of  three  figures) ;  since  the  square  of  the  figure 
standing  in  the  hundred^s  place  cannot  be  found  either  in  the 
last  period  (61),  or  in  the  Jast  but  one  (33),  it  must  be  found 
in  the  first  period. (5) ;  consequently  the  first  figure  of  the  root 
will  be  the  square  root  of  the  greatest  square  number  contained 
in  5  ;  and  this  number  is  4,  the  first  figure  of  the  root  will  be 
2.  The  remainder  of  the  operation  will  be  readily  understood 
by  comparing  the  steps  of  it  with  the  several  steps  of  the  pro- 
cess for  finding  the  square  root  of  (a  +  &4-c)2  (Art.  238) ;  for, 
having  subtracted  4  from  (5),  there  remains  1  ;  bring  down 
the  next  two  figures  (33),  and  the  dividend  is  133  ;  double  the 
first  figure  of  the  root  (2),  and  place  the  result  4  in  the  divisor  ; 
4  is  contained  in  13  three  times;  3  is  therefore  the  second 
figure  of  the  root ;  place  this  both  in  the  divisor  and  quotient, 
and  the  former  is  43  ;  multiply  by  3,  and  subtract  129,  the  re- 
mainder is  4  ;  to  which  bring  down  the  next  two  figures  (61), 
which  gives  461  for  a  dividend.  Lastly,  double  the  last  figure 
of  the  former  divisor,  and  it  becomes  46  ;  place  this  in  the  next 
divisor,  and  since  4  is  contained  in  4  once^  1  is  the  third  figure 
of  the  root ;  place  1  therefore  both  in  the  divisor  and  quotient ; 
multiply  and  subtract  as  before,  and  nothing  remains. 

247.  The  method  of  extracting  the  cube  root  of  numbers 
may  be  understood  by  comparing  the  process  for  extracting 
the  cube  root  of  (a-j-^+c)^,  (Art.  239),  with  the  following 
operations,  in  which  is  deduced  the  cube  root  of  the  number 
13997521. 


EVOLUTION.  195 


13997521(200+40+1 
o3=(200)3=8000000 


1st  remainder  5997521 


3a2=3x(200)2.=  divisor, 

•.•  3a2^>  =  3(200)2    X  40=4800000 

3a62=3x200x(40)2=  960000 

63=40x  40x40=     64000 


5824000 


2nd  remainder    173521 


3(a+&)2c=3(200  +  40)2x  1  =  172800 

3((2+i)c3=3(200  +  40)  X  1  =       720 

c3=lxlXl^  1 


173521 


3d  remainder  000000 


Omitting  the  superfluous  ciphers,  and  bringing  down  three 
figures  at  a  time,  the  operation  will  stand  thus  ; 

13997521)241 
23=  8 


5997 


300x2^X4=  4800 

30X2X42=     960 

43=       64 


5824 


173521 

300  X  (24)2x1  =  172800 

30x24x12=        720 

13=  1 


173521 


196  EVOLUTION. 

248.  These  operations  may  be  explained  in  the  following 
manner  ; 

I.  Since  the  cube  root  of  1000  is  10,  of  1000000  is  100, 
&c. ;  it  follows,  that  the  cube  root  of  a  number  less  than  1000 
will  consist  of  one  figure ;  of  a  number  between  1000  and 
1000000  oitvoo  figures,  &;c.  &c.  ;  if,  therefore,  the  given  num- 
ber be  divided  into  periods,  each  consisting  of  three  /figures, 
by  placing  a  dot  over  every  third  figure,  beginning  with  the 
units,  the  number  of  those  dots  will  show  the  number  of 
figures  of  which  the  cube  root  consists  ;  and  for  the  reason 
assigned  in  the  preceding  Article,  (respecting  the  fir^  figure 
of  the  square  root),  the  first  figure  of  the  root  will  be  the 
cube  root  of  the  greatest  cube  number  contained  in  the  first 
period. 

II.  Having  pointed  the  number,  we  find  that  its  cube  root 
consists  of  three  figures.  The  first  figure  is  the  cube  root  of 
the  greatest  cube  number  contained  in  13  ;  this  being  2,  the 
value  of  this  figure  is  200,  or  a =200,  consequently  0^= 
8000000  ;  subtract  this  number  from  13997521,  and  the  re- 
mainder is  5997521.  Find  the  value  of  ^x'^,  and  divide  this 
latter  number  by  it,  and  it  gives  40  for  the  value  of  a,  the  se- 
cond number  of  the  root ;  put  this  in  the  quotient,  and  then 
calculate  the  value  of  3aV}-\-^ah'^-\-P,  and  subtract  it,  and 
there  remains  173521.  Find  now  the  value  of  3x(«^-^)^ 
and  divide  173521  by  it,  and  it  gives  1  for  the  value  of  c,  the 
third  member  of  the  root ;  put  this  in  the  quotient,  and  then 
calculate  the  amount  of  3(a+^)2c4-3(a+Z>)c2-f-c^  which  sub- 
tract, and  nothing  remains. 

III.  In  reviewing  the  first  of  these  two  operations,  it  is 
evident  that  six  ciphers  might  have  been  rejected  in  the  va- 
lue of  a^,  and  three  in  the  value  oi^d^b-\-3ali^-^h'^,  without  af- 
fecting the  substance  of  the  operation  ;  having  therefore  sim- 
plified the  process  as  in  the  second  operation,  we  are  fur- 
nished with  the  following  rule,  for  extracting  the  cube  root  of 
numbers. 


RULE. 

249.  Point  off  every  third  figure,  beginning  with  the  units  ; 
find  the  greatest  cube  number  contained  in  the  first  period, 
and  place  the  cube  root  of  it  in  the  quotient.  Subtract  its 
cube  from  the  first  period,  and  bring  down  the  next  three 
figures  ;  divide  the  number  thus  brought  down  by  300  times 
the  square  of  the  first  figure  of  the  root,  and  it  will  give  the 
second  figure  ;  add  300  times  the  square  of  the  first  figure, 


•  EVOLUTION.  197 

30  times  the  product  of  the  first  and  second  figures,  and  the 
square  of  the  second  figure  together,  for  a  divisor  ;  then  mul- 
tiply this  divisor  by  the  second  figure,  and  subtract  the  result 
from  the  dividend,  and  then  bring  down  the  next  period,  and 
so  proceed  till  all  the  periods  are  brought  dovi^n. 

The  rules  for  extracting  the  higher  powers  of  numbers,  and 
of  compound  algebraic  quantities,  are  very  tedious,  and  of  no 
great  practical  utility. 

Examples  for  practice  in  the  Square  and  Cube  Roots  of 
Numbers. 


Ex.  1.  Required  the  square  root  of  106929. 
106929(327 
9 

62     169 
124 

647 

4529 
4529 

Ex.  2.  Required  the  cube 

)  root  of  48228544. 

48228544(364 
27 

3276(21228 
19656 

Divide  by  300X32=2700 

30x3x6=  540 

6X6=     36 

393136)  1572544 
1572544 

1st  Divisor  =3276 

Divide  by,  (36)2x300=388800 

30x36x4=     4320 

4x4=         16 


2d  Divisor  393135 
Ex.  3.  Required  the  square  root  of  152399025. 

Ans.  12345. 
Ex.  4.  Required  the  square  root  of  5499025. 

Ans.  2345. 
Ex.  5.  Required  the  cube  root  of  389017.  Ans.  73. 

Ex.  6.  Required  the  cube  root  of  1092727.  Ans.  103 

18*  ' 


198 


CHAPTER  VII.. 

ON 

IRRATIONAL  AND  IMAGINARY  QUANTITIES. 

§  I.    THEORY  OF  IRRATIONAL  QUANTITIES. 

250.  It  has  been  demonstrated  (Art.  231),  that  the  mih.  root 
of  a^f  the  exponent  p  of  the  power  being  exactly  divisible  by 

the  index  m  of  the  'root,  is  a'".  Now  in  case  that  the  expo- 
nent p  of  the  power  is  not  divisible  by  the  index  m  of  the  root 
to  be  extracted,  it  appears  very  natural  to  employ  still  the  same 
method  of  notation,  since  that  it  only  indicates  a  division  which 
cannot  be  performed  :  then  the  root  cannot  be  obtained,  but 
its  approximate  value  may  be  determined  to  any  degree  of  ex- 
actness. These  fractional  exponents  will  therefore  denote  im 
perfect  powers  with  respect  to  the  roots  to  be  extracted ;  and 
quantities,  having  fractional  exponents,  are  called  irrational 
quantities,  or  surds. 

It  may  be  observed  that  the  numerator  of  the  exponeit 
shows  the  power  to  which  the  quantity  is  to  be  raised,  and  the 

denominator  its  root.     Thus,  a"  is  the  wth  root  of  the  mih 

power  of  a,  and  is  usually  read  a  in  the  power  ( — j. 

251.  In  order  to  indicate  any  root  to  be  extracted,  the  ra- 
dical sign  y  is  used,  which  is  nothing  else  but  the  initial  of 
the  word  root,  deformed,  it  is  placed  over  the  power,  and  in  the 
opening  of  which  the  index  m  of  the  root  to  be  extracted  is 

written. 

p_ 

We  have  therefore  Y  a^  =a"'.  For  the  square  root,  the 
sign  -/  is  used  without  the  index  2  ;  thus,  the  square  root  of 
a^  is  written  •v^a^  as  has  been  already  observed,  (Art.  18). 

Quantities  having  the  radical  sign  ^  prefixed  to  them,  are 
called  radical  quantities:  thus,  J/ a,  -/^,  {/ c2,y/ a;"',&c,  are 
radical  quantities ;  they  are,  also,  commonly  called  Surds. 

252.  From  the  two  preceding  articles,  and  the  rules  given 
in  the  second  section  of  the  foregoing  Chapter,  we  shall,  in 
general,  have 


IRRATIONAL  QUANTITIES.  199 

p_       t-       L 
y  (a^.h''.c')  =  y  aPxyb''X'yc'z=a'^xb^Xe^\ 

f^la/'.h'' _y  {aK¥)  _y  ¥  x"y/ b^ _a^ Xh^ 

V7F~  ycd'  ~Yc'xyd''~  '-     '' 

Therefore,  ^  aH=y  a^X^  bz=aX^  b=ay  b  ; 

^ja^^c^^J/  a%^c'^_^  a^X^b'^xyc^ 

_a^by  c^^a^b  ^jc^ 
exy  xz        ex'y  xz' 

253.  Two  or  more  radical  quantities,  having  the  same  in- 
dex, are  said  to  be  of  the  same  denomination,  or  kind  ;  and  they 
are  of  different  denominations,  when  they  have  different  indices. 

In  this  last  case,  we  can  sometimes  bring  them  to  the  same 
denomination ;  this  is   what  takes  place  with  respect  to  the 

twofollowing,  yaWar\dya%^  =  a^xb*=a^  .  b^-^a^^b^ 
=  -y/a^b"^.  In  like  manner,  the  radical  quantities  ^  2a^b  and 
y  160^6,  may  be  reduced  to  other  equivalent  ones,  having  the 
same  radical  quantity  ;  thus,  ^  2a^b=zy  a^  Xy^  2b=a'^  y  2b, 
and  y  I6a^=zy  8a^  .  2b=y  S.^aK^  2bz=2ay  2b  ;  where 
the  radical  factor  y  2b  is  common  to  both. 

254.  The  addition  and  subtraction  of  radical  quantities  can 
in  general  be  only  indicated  : 

Thus,  y  d^  added  to,  or  subtracted  from  yfb,  is  written  -y/h 
■^y  a^,  and  no  farther  reduction  can  be  made,  unless  we  as- 
sign numeral  values  to  a  and  b.  But  the  sum  of  -y/a^,  -y/a^b, 
and  ^^a^b  is  =a^/b-\-a^/b-{■2a^b—^a^/b^,  Sy  ab—^  ab 
—2yab',  and  ^/ ab'^-^y  a^'^—byf  a^aby  a^^^b^ a^ab^/ a 
—  {b-^ab)ya. 

255.  Hence  we  may  conclude,  that  the  addition  and  sub- 
traction of  radical  quantities,  having  the  same  radical  part,  are 
performed  like  rational  quantities. 

Radical  quantities  are  said  to  have  the  same  radical  part 
when  like  quantities  are  placed  under  the  safhe  radical  sign  ; 
in  which  case  radical  quantities  are  similar  or  like.  It  is  some- 
times necessary  to  simplify  the  radical  quantities,  (Art.  252), 
in  order  to  discover  this  similitude,  and  it  is  independent  ot 
the  coefficients. 

Thus,  for  example,  the  radical  quantities  35  y/  2a^b'^,  Sa^ 
2a^b^,  and  ^laby  2a%'^,  become,  by  reduction,  3o6^  2^262, 
Saby  2a%'^,  and  —laby  2aW  ;  which  are  similar  quantities, 
and  their  sum  is  =4^6^/  2a262. 


200  IRRATIONAL  QUANTITIES. 

256.  We  have  demonstrated,  (Art.  252),  this  formula, 
v^  aJ'Vc^—y  dPy."^  V'y.'y  c^  \  from  which  the  rule  for  the 
multiplication  of  radical  quantities,  under  the  same  radical 
sign,  may  be  easily  deduced. 

257.  Let  us  pass  to  radical  quantities  with  different  indices, 

and  suppose  that  we  had  to  find,  for  instance,  the  product  of 

±         ± 
'y  aPhy%/  h\  or  that  of  oT  by  5  *"' :  we  can  bring  this  case  to 

the  preceding,  by  reducing  to  the  same  denominator,  (Art. 

152),  the  fractions^  and^  ;  and  we  shall  have  V  a^X  V  Z>* 

'  mm 

p  1-       f""' .        ?"* 
=  a"^6"''=a'""-'  X  b"""'="""-l^  a'""'  X"™V'  b'"''=z"""y  oT'b''"'. 

258.  The  rule  for  dividing  two  radical  quantities  of  the 
same  kind,  may  be  read  in  this  formula  (Art.  233). 

y  a''__"'jaP 

and  it  only  remains  to   extend  it  to  two  radical  quantities  of 
different  denominations. 

Let  therefore  y  a''  be  divided  by  "^  b''  :  by  passing  from 
radical  signs  to  fractional  exponents,  we  have 

y  aP      a'"      a"""'     ^^y  a'""' _"""' /a'"'" 

We  may  likewise  suppose,  under  the  radical  signs,  any 
number  of  factors  whatever,  and  it  shall  be  easy  to  assign  the 
quotient,  (Art.  252). 

Let  now  a  =  6  in  the  formula 

y  aJ'xyb''  =  y  aP  .  b"] 
it  becomes,  by  passing  from  radical  signs  to  fractional  expo- 
nents, 

a™  X  a'"  =y  a^^^=a  ""  =a'^  "» . 
Therefore  the  rule  demonstrated  (Art.  71),  with  regard  to 
whole  positive  exponents,  extends  to  fractional  exponents. 

259.  In  the    same   hypotheses  5= a,  the  quotient  ^j^be- 


p_ 
comes 


„         m  l^p  p-1  p-q_ 

-=^-^=yaP-''=ra'^=a'^     -  ; 


or    . 
another  extension  of  the  rule  given  (Art.  86),  to  fractional 
positive  exponents. 


IRRATIONAL  QUANTITIES.  201 

260.  We  may,  in  the  preceding  formula,  suppose  p=o  ;  and 

1      ±  1       -i. 

it  becomes,  (since  a'"=a'"=a''=l)—=a  •",  a   transformation 

or 

demonstrated,  (Art.  86)  in  the  case  of  whole  exponents,  and 
which  still  takes  place  when  the  exponents  are  fractional. 

261.  If  we  now  admit  the  two  equalities, 

1      -?-      1         -^ 
p_     a    ,      ,_— a      J 

a™  or 

and  if  we  multiply  them  member  by  member,  we  shall  hare 
the  equal  products, 
111  1      i      i_L 

~l^~i— T^'    or  a^Xa'^o"*    "•* 
a  *"     oT     a""  ^ '" 

It  appears  therefore  evident,  that  exponentials  with  frac- 
tional negative  exponents,  follow  the  same  rule  in  their  mul- 
tiplication, as  those  with  whole  positive  exponents. 

__£  _-? 

[  262.    The  division  of  oT  by   a",  gives  for  the  quotient, 

a"*      or 

Now  the  exponent  of  the  quotient,  namely-^ 4.  J^,  is  the  expo* 

nent  of  the  dividend,  minus  that  of  the  divisor,  which  is  still 
a  generality  of  the  rule  (Art.  86),  relative  to  the  division  of 
exponentials. 

263.  The  rules  that  have  been  demonstrated  in  the  pre- 
ceding articles  may  be  extended  to  radical  quantities  having 

irrational  exponents :  For  instance,  — 7—  7— rr,  &c.  since  that 

ay  Z,  0  yf  6 

the  roots  of  -^1  and  -v/S  might  be  obtained  with  a  sufficient 
degree  of  approximation,  and  such  that  the  error  may  be  ne- 
glected ;  so  that  these  exponents  shall  be  terminated  decimal 
fractions,  which  can  be  always  replaced  by  ordinary  fractions. 

264.  The  formation  of  the  powers  of  radical  quantities,  is 
nothing  else  but  the  multiplication  of  a  number  of  radical 
quantities  of  the  same  denomination,  marked  by  the  degree 
of  the  power  ;  so  that  it  is  sufficient  to  raise  the  quantity 

^       under  the  radical  sign  to  the  proposed  power,  and  afterwards 


202  IRRATIONAL  QUANTITIES 

to  affect  this  power  with  the  common  radical  sign.  If  the  in- 
dex of  the  radical  sign  is  divisible  by  the  exponent  of  the 
power  in  question,  the  operation  then  is  performed  by  dividing 
that  index  by  the  exponent  of  the  power.  Let  us  give  two 
examples  for  these  two  cases,  (^  a''b'')'=y  aP'b'"  ;  Cy  a^^)' 
=y  aPb". 

265.  If  the  exponent  of  the  power  is  equal  to  the  index  of 
the  radical  sign,  the  power  is  the  quantity  under  the  radical 
sign.  In  fact,  the  indication  y  a'',  shows  that  a^  is  the  mth. 
power  of  a  certain  number  y  a",  which  we  can  always  assign, 
either  rigorously,  or  by  an  approximation,  so  that  the  mth. 
power  of  y  a''  is  a^.  In  like  manner,  the  square  of  -y/a  is 
a ;  the  cube  of  ^  a  is  a  ;  the  5th  power  of  ^  (— «^)  is  ~d^  ; 
and  so  on. 

266.  A  rational  quantity  may  he  reduced  to  the  form  of  a  given 
surd,  hy  raising  it  to  the  power  whose  root  the  surd  expresses,  and 
prefixing  the  radical  sign.    Thus  a^=z^  a^^=zy  a^=^  a^,  &c. 

m 

and  a-\-x=^{a-\-x)"'.  In  the  same  manner,  the  form  of  any 
radical  quantity  may  be  altered  ;  thus,  '^{a-\-x)=iy  (a+^)^  = 

^  {afxY,  &LC.  or  (a4-a;)2  —  («-f  a;)4  —  («-f  a;)6,  Slc.  Since  the 
quantities  are  here  raised  to  certain  powers,  and  the  roots  of 
those  powers  are  again  taken  ;  therefore  the  values  of  the 
quantities  are  not  altered.  Also,  the  coefficient  of  a  surd  may  he 
introduced  under  the  radical  sign,  hy  first  reducing  it  to  the  form 
of  the  surd,  and  then  multiplying  as  in  (Art.  257).     Thus,  a^x 

—  ^/a^X^/xz=z^a'^x',    6^2  =  ^36  Xy^2  =  'v/'''2;   and  ir(2a 

-.ir)^=(a;2)ix(2a-xP=  y/ (20x^-0?). 

267.  Conversely,  any  quantity  may  be  made  the  coefficient  of  a 
surd,  if  every  part  under  the  sign  be  divided  by  this  quantity, 
raised  to  the  power  whose  root  the  sign  expresses.     Thus,   Vl"^ 

—  a2^)=  ■^/a'^X^/(a—x)=a^{a—x)\  ^60=  -v/(4Xl5)=i 
y'4x  •v/15=2-v/15  ;  and  'y  (a'""  — a-^a:")  =y  [a"'X  («"— a^")] 
=y  a'»  X"/  (a"— x")  =  ay  (a"— a;"). 

268.  Let  us  pass  to  the  extraction  of  roots  of  radical  quan- 
tities, and  let  the  mth  root  of  y  a*  be  required,  which  we  in- 
dicate thus,  y  y  a*.  We  shall  put  y  y/a^  —  x,  or  y  a'=.x,  by 
making  y  a*=ia'.  Involving  both  sides  to  the  power  m,  we 
find  a'  or  y  a*=x'",  raising  again  to  the  power  n,  we  obtain 
a'=x"'".  If  the  mnih  root  of  both  sides  be  extracted,  we  have 
another  enunciation  of  x ;   namely, 

'"ya'=:x=yya'. 


IRRATIONAL  QUANTITIES.  203 

We  shall  find,  by  a  like  calculation, 

And,  in  fact,  we  make  lst,y  {/{/  a'=a',  whence  y  a'=:x,  and 
a'=y(/y  a'=:a;'";  2d,  by  putting  {/{/ a' =  a",  whence  ^  a" 
=  07'",  and  a^'z^a;'"'' ;  3d,  making {/  a*=ia"',  whence  {/  a^"=a;"'", 

and  a"'=y  a'=o(r"P  ;  and  finally  a'=a;'^w^  .-.  a:=     ya'. 
Thus,  for  example,  the  12th  root  of  the  number  a  can  be  trans- 
formed into  ^  ^  y/  «, 

169.  It  is  to  be  observed,  that  radical  quantities  or  surds, 
when  properly  reduced,  are  subject  to  all  the  ordinary  rules 
of  arithmetic.  This  is  what  appears  evident  from  the  preced- 
ing considerations.  It  may  be  likewise  remarked,  that,  in  the 
calculations  of  surds,  fractional  exponents  are  frequently  more 
convenient  than  radical  signs. 

§  II.    REDUCTION  OF  RADICAL  QUANTITIES  OR  SURDS. 

CASE  I. 
To  reduce  a  rational  quantity  to  the  form  of  a  given  Surd. 

RULE. 

270.  Involve  the  given  quantity  to  the  power  whose  root 
the  surd  expresses  ;  and  over  this  power  place  the  radical  sign, 
or  proper  exponent,  and  it  will  be  of  the  form  required. 

Ex.  1 .  Reduce  a  to  the  form  of  the  cube  root. 
Here,  the  given  quantity  a  raised  to  the  third  power  is  a^, 
and  prefixing  the  sign  ^  ,  or  placing  the  fractional  exponent 

(^)  over  it,  we  have  a=^  a^={a^)'^  (Art.  251). 

271 .  A  rational  coefficient  may,  in  like  manner,  be  reduced 
to  the  form  of  the  surd  to  which  it  is  joined  ;  by  raising  it  to 
the  power  denoted  by  the  index  of  the  radical  sign. 

Ex.  2.  Let  5-v/a=y'25X'/«  =  V25a. 
Ex.  3.  Reduce  — da'^b  to  the  form  of  the  cube  root. 
Here,  {  —  3aHy  =  -27a^P  ;   .-.  — ^  27a6J3  is  the  surd  re- 
quired. 

Ex.  4.  Reduce  —4x1/  to  the  form  of  the  square  root. 
Here,  (—4xy)'^=zl6xY  ;  .-.  —  4a;y=  — Vl^^V- 
Ex.  5.  Reduce  4a;  to  the  form  of  the  cube  root. 

^  1 

Ans.  (ia;3)^. 


204  IRRATIONAL  QUANTITIES. 

Ex.  6.  Reduce  a-\-z  Xo  the  form  of  the  square  root. 

Ans.  {a'^-^2az+z'^)^ 
i. 
Ex.  7.  Reduce  4a;*  to  the  form  of  the  cube  root, 

Ans.  (^  64a;* )  or  (64a;*)^. 

Ex.  8.  Reduce  —x^y^  to  the  form  of  the  square  root, 

Ans.  — -y/ocy. 
Ex.  9.  Reduce  —ah  to  the  form  of  the  square  root. 

Ans.  — ■y/oP-h'^, 

CASE  II. 

To  reduce  Surds  of  different  indices  to  other  equivalent 
ones,  having  a  common  index. 

RULE. 

272.  Reduce  the  indices  of  the  given  quantities  to  fractions 
having  a  common  denominator,  and  involve  each  of  them  to 
the  power  denoted  by  its  numerator  ;  then  1  set  over  the  com- 
mon denominator  will  form  the  common  index. 

Or,  if  the  common  index  be  given,  divide  the  indices  of  the 
quantities  by  the  given  index,  and  the  quotients  will  be  the 
new  indices  for  those  quantities.  Then  over  the  said  quantities, 
with  their  new  indices,  set  the  given  index,  and  they  will  make 
the  equivalent  quantities  sought. 

Ex.  1.  Reduce  ^/a  and  ^5  to  surds  of  the  same  radical 
sign. 

Here,  -yjazzza^,  and  ^  h  —  h^ .  Now,  the  fractions  \  and  \ 
reduced  to  the  least  common  denominator,  are  \  and  | ; 

.-.  ^=:a^^{a?f^ya^,  and  h^ =^  =:{h''f  ^^  V^ . 
Consequently  ^   a^  and  ^  h^  are  the  surds  required 
Ex.  2.  Reduce  -y/a  and  ^  x  to  surds  of  the  same  radical 

sign  ^  ,  or  to  the  common  index  4. 
1  X 

(Art.  251),  ^Ja—c?,  and  y  a;=a:*  ;  then  ^4-^=^X6  =  3  ; 

and  i-r- J=i X  6=3| ;  .-. |/  a^  and  ^  o? ,  or  {a^^  and  {i^f,  are 
the  quantities  required. 

Ex.  3.  Reduce  a^  and  i^  to  the  same  radical  sign  ^  . 

Ans.  y  a®,  and  ^  b^» 


IRRATIONAL   QUANTITIES.  205 

Ex.  4.   Reduce  a*  and  x^  to  surds  of  the  same  radical  sign 

Ans.  ^^  x^  and  ^ ^  aj*. 

Ex.  5.   Reduce  y  a  and  v^  y  to  surds  of  the  same  radical 

sign.  Ans.  ""^  a'"  and  ""^  y". 

2.  1 

Ex.  6.   Reduce  a^  and  i*  to  surds  of  the  same  radical  sign. 

Ans.  i^a^  and^y  R 
Ex.  7.  Reduce  3^  2  and  2-y/5  to  the  same  radical  sign. 

Ans.  3^4  and  2^  125. 
Ex.  8.   Reduce  \/  xy  and  ^/  ax  to  the  same  radical  sign. 

Ans  ^^  x'^y^  and  ^^.  a^x"^, 

CASE   III. 

To  reduce  radical  Quantities  or  Surds,  to  their  most  simple 
forms. 

RULE. 

273.  Resolve  the  given  number,  or  quantity,  under  the  ra- 
dical sign,  if  possible,  into  two  factors,  so  that  one  of  them 
may  be  a  perfect  power  ;  then  extract  the  root  of  that  power, 
and  prefix  it,  as  a  coefficient  to  the  irrational  part. 

Ex  1.  Reduce  ■\/a'^b  to  its  most  simple  form. 

Here  Va25_yfo2xy'^>=aXV'^=«'/*• 
Ex.  2.  Reduce  Y  ^"^  ^^  i^^  most  simple  form. 

ni 

Uerey  a'^x^y  a'"X'y  x=za^xy  x=zaxy  X. 
Ex.  3.  Reduce  ■v/'^2  to  its  most  simple  form. 
Here  ^72  =  V(36  X2)=r -/SSx  ^2=6^2. 

274.  When  the  radical  quantity  has  a  rational  coefficient 
prefixed  to  it ;  that  coefficient  must  be  multiplied  by  the  root 
of  the  factor  above  mentioned  ;  and  then  proceed  as  before. 

Ex.  4.  Reduce  5^  24  to  its  simplest  form.# 
Here  5^  24  =:  5y  (8  x  3)  =  5^/  8  x  V  3^  5  X 2  X  3/  3  = 
10V3. 

Ex.  5.   Reduce   -ya^hc  and  -^^^d^x  to  their  most  simple 

form.  Ans.  a'^-^hc  and  '7a^/2x. 

Ex.  6.   Reduce  1/  243  and  V  96  to  their  most  simple  form. 

Ans.  3y  3  and  2^  3. 
Ex.  7.  Reduce  ?/  (a^-ira^"^)  to  its  most  simple  form. 

Ans.  aV(l  +  ^'). 

Ex.8.  Reduce     /( ^ J  to  its  most  simple 

a—2h   ,  , 
form.  Ans.  yab, 

19 


206  IRRATIONAL  QUANTITIES. 

Ex.9.  Reduce  {a-{-b)\/ [(a—bYxx"^]  to  its  most  simple 
form.  Ans.  (fl2_i2)3^^2, 

275.  If  the  quantiti;  under  the  radical  sign  he  a  fraction,  it 
may  he  reduced  to  a  ivhole  quantity^  thus  : 

Multiply  both  the  numerator  and  denominator  by  such  a 
quantity  as  will  make  the  denominator  a  complete  power  cor- 
responding to  the  root ;  then  extract  the  root  of  the  fraction 
whose  rmmeralor  and  denominator  are  complete  powers,  and 
take  it  frorn  under  the  radical  sign. 

c  a'- 

Ex.  1.  Reduce  -  X  i/-T- lo   aa    intes^ral  surd  in  its  most 
d      ^  b  ^ 

simple  form. 

Tx  c     ,(P      c     ,a'^b     c     ,d^      ,      c      a     ,.      ca     ,, 

Ex.  2.  Reduce  i^  ij  to  an  integral  surd  in  its  simplest  form. 

/  H  V  2  \  v>  V  ^2 

nere,  ^  /  -gr— ^  /  \^27  x  3 j  ~  a  v  2T  ^  /  a  —  »  X  ^v  — ^j— 

Ex.  3.  Reduce  J-v/f  ^"^  ^^  integral  surd  in  its  most  simple 
form.  An&.  ^^g-y/K. 

b  c^ 

Ex.  4.  Reduce  x-y/  -  and  «?/  —  to  inteo-ral  surds  in  their 

y  ^   a  ^ 

most  simple  forna.  Ans.  -  -/^^  and  ^  e^a^, 

Ex.  5.  Reduce  ^  i  and  f -/J  icy  integral  surds  in  their  most 
simple  form.  *"  Ans.  ^\^  21  and  3'\/2. 

Ex.  6.  Reduce  ?/  -— -  and  -\/  —--  t€>  their  most  simple  form. 
^   125         ^  Sx*  ^ 

•  Ans.  "V  2  and  — -r\/2a. 

5  *^        ^     4a;2  '^ 

276.  The  utility  of  reducing  surds  to  their  most  simple  forms, 
especially  when  the  surd  part  is  fractional,  will  be  readily  per- 
ceived from  the  3d  example  above  given,  where  it  is  found  that 
f -v/f^-g^V^M,  in  which  case  it  is  only  necessary  to  extract, 
the  square  root  of  the  whole  number  14,  (or  to  find  it  in  some  of 
the  tables  that  have  been  calculated  for  that  purpose),  and  then 
multiply  it  by  ^ ;  whereas  we  must,  otherwise,  have  first  divid- 
ed the  numerator  by  the  denominator,  and  then  have  found  the 
root  of  the  quotient,  for  the  surd  part  ;  or  else  have  determined 
the  root  of  both  the  numerator  and  denominator,  artd  then  divide 
the  one  by  the  other ;  which  are  each  of  them  troublesome  pro- 


IRRATIONAL  QUANTITIES.  207 

cesses ;  and  the  labour  would  be  much  greater  for  the  cube 
and  other  higher  roots. 

277.  There  are  other  cases  of  reducing  algebraic  Surds  to 
simpler  forms,  that  are  practised  on  several  occasions  ;  for  in- 
stance, to  reduce  a  fraction  whose  denominator  is  irrational,  to 
another  that  shall  have  a  rational  denominator.  But,  as  this 
kind  of  reduction  requires  some  farther  elucidation,  it  shall  be 
treated  of  in  one  of  the  following  sections. 

§  III.    APPLICATION    OF    THE    FUNDAMENTAL  RULES  OF  ARITH- 
METIC TO  SURD  (QUANTITIES. 

CASE  I. 

To  add  or  subtract  Surd  Qiuzniities, 

RULE. 

278.  Reduce  the  radical  parts  to  their  simplest  terms,  as  in 
the  last  case  of  the  preceding  section  ;  then,  if  they  are  similar, 
annex  the  common  surd  part  to  the  sum,  or  difference  of  the 
rational  parts,  and  it  will  give  the  sum,  or  difference  required. 

Ex.   1.  Add  4  v^ a?,  -/a;,  and  5 -y/ a;  together. 
Here  the  radical  parts  are  already  in  their  simplest  terras, 
and  the  surd  part  the  same  in  each  of  them;  .'.4:-)/x-{--\/x 
■i-5i/x={44-l-\-^)X  y'jr=:10y'a;  the  sum  required. 

Ex.  2.  Find  the  sum  and  difference  of  '\/\&a'^x  and  -y/Aa^x. 
'\/lQa^x=:-y/\Qa'^X  -y/xzizAa^Xf 
and  -/4a%=  ■y/i.a^  X  ^xz=z2a-}/x  ; 
.'.  the  57iwi  =:(4<2-j-2a)  X -y/jp^Say^a? ; 
and  the  difference  ={4« — 2a)  X  ^/x\=:2a^/x. 
Ex.  3.  Find  the  sum  and  difference  of  ^  108  and  9^/  32. 
Here^l08=:^27X^4  =  3x|/4=   3^/4, 
and  9^  32:r39y  8X^  4  =  18  X^  4  =  18^  4, 
the  5wm  =(18+3)x^4=2l|/4; 
and  the  difference  =(18  — 3)  X'V  4=15^/  4. 

279.  If  the  surd  part  be  not  the  same  in  each  of  the  quan- 
tities, after  having  reduced  the  radical  parts  to  their  simplest 
terms,  it  is  evident  that  the  addition  or  subtraction  of  such 
quantities  can  only  be  indicated  by  placing  the  signs  +  or  ^ 
between  them. 

Ex.  4.  Find  the  sum  and  difference  of  3^  a%  and  hs/c^d. 
Here  3J/  a'^h=^'iy  a^  X  3/  ^>=3«  X^  hr^Za]/  h, 
and  hyj c^d—hA/ c'^  X  -y/d—hc X  -y/d^hc yfd ; 
the  sum  =z3al/  b-\-bc-y/d- 
and  the  diff'erence  z=3a^  h^hc-^/d. 


208  IRRATIONAL  QUANTITIES. 

Ex.  5.  Find  the  sum  and  difference  of  -/^^y  and  -/J. 

Ans.  The  sum  =z:^j^6,  and  difference  =Jg.y^6. 

Ex.  6.  Find  the  sum  and  difference  of  ■\/27a^x  and  -/3a*a:. 

Ans.  The  ^wm  =4a'^^3x,  and  difference  —2a?-^3x. 

Ex.  7.  Find  the  ^w»i  and  difference  of  J -/a^ft  and  ^^-/^a;*. 

A         rr^u                    /2a:24-3a\     ..        ^    ^.^             /'2x'^2a\ 
Ans.  The   ^wm  =» — — jy^,  and   difference  i ) 

Ex.  8.  Required  the  sum  and  difference  of  3^/  62.5  and 
2|/  135. 

Ans.  The  ^wwi  =21|/  5,  and  difference  =9^  5. 

Ex.  9.  Required  the  ^mw  and  difference  of  y  a^i^  ^nd  ^  a;^^^. 

Ans.  The  5«m  z=.a^/ab-\-xy  x^y"^,  and  difference  z^a-^ah^ 
x^  x^y^, 

CASE  II. 

To  multiply  or  divide  Surd  Quantities. 

RULE. 

280.  Reduce  them  to  equivalent  ones  of  the  same  deno- 
mination, and  then  multiply  or  divide  both  the  rational  and  the 
irrational  parts  by  each  other  respectively. 

The  product  or  quotient  of  the  irrational  parts  may  be  re- 
duced to  the  most  simple  form,  by  the  last  case  in  the  preced- 
ing section. 

1  jL 

Ex.  1.  Multiply  -y/a  by  ^  b,  or  a^  by  b^. 

The  fractions  J  and  ^,  reduced  to  a  common  denominator^ 
are  J  and  ^. 

.-.  a^=^a^=^  «3  ;  and  b^=b^=^  b\ 
Hence  V<zX^  ^>*=^  a^x^/ J2_6/ ^352, 

Ex.  2.  Multiply  2^3  by  3iJ/  4. 

3 
By  reduction,  2^/'6=2  x  3«  =2  x «/  33=2^  27 ; 

and  3^  4^:3  X  4^=3^42=^3^  16. 
.-.  2 V3 X  3^  4=2?/  27  X  3^  16=:6f/  432. 
Ex.  3.  Divide  8^  512  by  4^/  2. 

Here  8-^4=2,  and  ^  512-^-^  2=^  256=4^  4. 
.-.  83/  512-r43/  2=2  X43/  4  =  8^  4. 
Ex.  4.  Divide  2  3/  be  by  3  Vac. 

Now  2^  &c=2  X  (icf =2  X  {hcf  =2^  ^2^2 


IRRATIONAL  QUANTITIES. 


209 


1  ^ 

and  3  7ac=3  X  (ac)2  =zz3  X  (ac)^  =  3^  aV  . 

•'•  37^~3^  V^3c"^~3V^~3V   a6c6  ~3acV     ""  "^  * 

281.  If  two  surds  have  the  same  rational  quaMity  under  the 
radical  signs,  their  product^  or  quotient,  is  obtained  by  making 
the  sum,  or  difference,  of  the  indices,  the  index  of  that  quantity. 


Ex.  5.  Multiply  ^  a^  hy  ^  a^  or  a^  by  a^ . 

Here,  a^  X  cP =^  ^=a^=a^.  Ov^a^X^  0^=^  (a^Xa^) 


=  ?/  a^^a"^,  as  before. 

Ex.  6.   Divide  y  a^  by  |/  a\  or  a*  by  «=^ 

3  4  ji_4  9        16 

Here,  aT-ra'  —  "    »-- .TJ-W 


282.  If  compound  surds  are  to  be  multiplied,  or  divided,  by 
each  other,  the  operation  is  usually  performed  as  in  the  multi- 
plication, or  division  of  compound  algebraic  quantities.  It  fre- 
quently happens  that  the  division  of  compound  surds  can  only 
be  indicated. 

Ex.  7.  Multiply  ^S—ya'^hy^S  +  ya. 

-/S-^^' ^  Since   V-'^X^  3=3^X  3^  = 
ys-h^a    iy  (3^X3^)=:^  (27X9)=^ 

. n/  243 

y243-y{3«2) 

+y{27a^)-a 

Product  =^  243-y  (Sa^)^^/  27a^-a. 

Ex.  8.  Divide  ^b'^ca-{-'\/a'^b  — be—  ^abc  by  ^bc+  y/a. 


•\/b'^ca  -{--yj  a-b  —  bc  —  -^abc 
^Jb'^ca-{^^/d^b 

— be — -y/abc 
— be  —  ^/abc 


Ex. 
Ex. 
Ex. 
Ex. 
Ex. 
Ex. 


^bc-\-  ^a 

Quot.  — -v/6a—  'Jbc. 


9.  Multiply  3/  15  by  -/ 10. 

10.  Multiply  13/ 6  by|3/  18. 

11.  Multiply  3/18  by  ^4. 

12.  -Multiply  \y  6  by  -f^y  9. 

13.  Divide  4-/50  by  2'/5. 

14.  Divide  IVI  by  W\. 
Ex.  15.  Divide  |/ 0^(^352  by  yrf: 

19* 


Ans.  y  225000. 

Ans.  ^  4. 

Ans.  23/  9. 

Ans.  -^y  2. 

Ans.  2v^l0. 

Ans  f -y/lO. 

Ans.^ai. 


210  IRRATIONAL  QUANTITIES. 

a     i  1     JL  IJL     2 

Ex.  16.  Multiply  a^  x^  by  a*  x^.  Ans.  a^^  x^ 

5. 

Ex.  17.  Multiply  y  aH^c^  by  y  a^PcK  Ans.  a^^c^. 

Ex.  18.  Divide  (a^+Pfhy  (a'^+b^f 

Ans.(/(a*+&^). 

Ex    19.  Multiply  4+2 -v/a  by  2-^2.  Ans.  4. 

Ex.  20.  Multiply  V(«  -  V(*  —  V^))  by  V(«  +  Vl*— 

V3)).  Ans.  V(a2-6+'/3). 

Ex.  21.  Divide  a35_aJ2c  by  a^-\-ay^bc. 

Ans.  ab—b^bc. 
Ex.  22.  Divide  a*+a;*  by  a2+aa;-/2  +  a:2 

Ans.  a^ — ax'^2-\-x'^. 

283.  It  is  proper  to  observe,  sin-ce  the  powers  and  roots  of 
quantities  may  be  expressed  by  negative  exponents,  that  any 
quantity  may  be  removed  from  the  denominator  of  a  fraction  into 
the  numerator  ;  and  the  contrary,  by  changing  the  sign  of  its  index 
or  exponent ;  which  transformation  is  of  frequent  occurrence  in 
several  analytical  calculations. 

1  a2 

Ex.  1.  Thus,  (since  — =5-^),  —  may  be    expressed  by 

1  a^         \ 

a2j-3  ;  and  (since  a^= — -),  we  have  7-^=71 — "o- 

a%^ 
Ex.  2.  The  quantity  —^  may  be  expressed  by  a^Pc-*e-^, 
c  e 

i   2 

^2^3 

Ex.  3.  Let  the  denominator  of  — r^r-  be  removed  into  the 

c  b^ 

1.  Z 
numerator.  Ans.  a^x^c-'^b—^. 

Ex.  4.  Let  the  numerator  of  -7—  be  removed  into  the  deno- 

o 

minator.  Ans. 


a-^x-^b 


Ex.  5.  Let  x^y^a^  be  expressed  with  a  negative  exponent. 

Ans. 


v-^ir-^a  * 


IRRATIOxNAL  QUANTITIES.  211 

CASE  III. 
To  involve  or  raise  Surd  Quantities  to  any  power 

RULE. 

284.  Involve  the  rational  part  into  the  proposed  power,  then 
multiply  the  fractional  exponents  of  the  surd  part  by  the  index 
of  that  power,  and  annex  it  to  the  power  of  the  rational  part, 
and  the  result  will  be  the  power  required. 

Compound  surds  are  involved  as  integers,  observing  the 
rule  of  multiplication  of  simple  radical  quantities. 
Ex.  1.  What  is  the  square  of  2^ a  1 

The  square  of  2^a—(2a^f=2'^  X  a^'    =4a. 
Ex.  2.  What  is  the  cube  of  ^  (a^—h'^-\-^/^)  ? 

The  cube  of  ^  {a^-h'^-\-^^)  =  (a'^-h'^^-^/^f^=a^^b'^ 
+  V3. 

285.  Cor.  Hence,  if  the  quantities  are  to  be  involved  to  a 
power  denoted  by  the  index  of  the  surd  root,  the  power  re- 
quired is  formed  by  taking  away  the  radical  sign,  as  has  be^a 
already  observed. 

Ex.  3.  What  is  the  cube  of  ^  y/2ax  1 

"  1.3  3 

Here  (\Y=^,  and  {^2axf=i{2axY'   z={2axY 

=z(2ax)x{2axy^  ;   .-.  \x2axX{2axY  = 
\ax^/2ax  is  the  power  required. 
Ex.  4.  It  is  required  to  find  the  square  of  -y/a—  V^- 

'y/a  —  ^/b 
-y/a — ^/b 


a  —  -y/ab 
—  \/ab-\-b 

The  square  a~~2-\/ab-\-h. 


Ex.  5.  It  is  required  to  find  the  square  of  3  ^Z  3. 

Ans.  93/9 
Ex.  6.  Find  the  cube  of  y/a.  Ans.  a^^/a 

Ex.  7.  Find  the  4th  power  of  —  ^  a-.  Ans.  a^^/  ^2 

Ex.  8.  Find  the  5th  power  of  —\^ab.  Ans.  —ab 

Ex.  9.  Required  the  cube  of  a —  -y/b. 

Kl\s.a^—^a^^/b-\■^ab—h^b 


212  IRRATIONAL  QUANTITIES. 

Ex.  10.  Required  the  square  of  3+  V^* 

Ans.  14+6V5. 
Ex.  11.  Required  the  cube  of  —  ^  W^~  V^^)- 

Ans.  -y/bc—  'sja. 

CASE  IV. 
To  evolve  or  extract  the  Roots  of  Surd  Quantities. 

RULE. 

286.  Divide  the  index  of  the  irrational  part  by  the  index  of 
the  root  to  be  extracted  ;  then  annex  the  result  to  the  proper 
root  of  the  rational  part,  and  they  will  give  the  root  required. 

If  it  be  a  compound  surd  quantity,  its  root,  if  it  admits  of 
any,  may  be  found,  as  in  Evolution.  And  if  no  such  root  can 
be  found,  prefix  the  radical  sign,  which  indicates  the  root  to 
be  extracted. 

Ex.  1.  What  is  the  square  root  of  81  ^/a  ? 

Here  -/81=9,  and  the  square  root  of  -y/a  or  a^z=a^-^2  = 

Jx^=ai^\/a  ;  .-.  '/(81/a)  =  9V  a,  or  9 A 
Ex.  2.  What  is  the  square  root  of  a^ — 6a-\/b-\-9b. 

a^—6a-^b  +  9b{a—3^/b 


2a-3y/b)-6a^/b+9b 
-6aV^  +  9i 


Ex.  3.  Find  the  square  root  of  9^  3.  Ans.  3  V  3. 

Ex.  4.  Find  the  4th  root  of  |}^  a^.  Ans.  f  (/  c. 


3 


Ex.  5.  Find  the  cube  root  of  (5a'^—3o^)^. 

Ans.  ^(5a'^-3x^). 
Ex.  6.  Required  the  cube  root  of  ^a^.  Ans.  ^a^  b. 

Ex.  7.  What  is  the  fifth  root  of  32^  a^*  ?  Ans.  2  3/  x. 

Ex.  8.  What  is  the  4th  root  of  16a^  x  ?         Ans.  2^  a*x. 
Ex.  9.  What  is  the  nth  root  of  7/  a''x^  ? 

LI. 
Ans.  a'^x^' 

Ex.  10.  It  is  required  to  find  the  cube  root  of  a^—3a^^x-{- 
Sax—x-^x.  Ans.  a—^x. 


IRRATIONAL  QUANTITIES.  213 

§  IV.  METHOD  OF  REDUCING  A  FRACTION,  WHOSE  DENOMI- 
NATOR IS  A  SIMPLE  OR  A  BINOMIAL  SURD,  TO  ANOTHER  THAT 
SHALL  HAVE  A  RATIONAL  DENOMINATOR. 

287.  A  fraction,  whose  denominator  is  a  simple  surd,  is  of 

the  form  - —  ;  where  x  may  represent  any  rational  quantities 
y  X 

whatever,  either  simple  or  compound  ;  thus, 

be  a  c — d        „ 

are  fractions,  whose  denominators  are  simple  surd  quantities. 

288.  It  is  evident  that,  if  a  surd  of  the  form  y  x  he  multi- 
plied by  y  a?"-^,  the  product  shall  be  rational  ;  since  y  xX 
y  x"—^=y  (xxx'^~^)=y  x"=x  ;  in  like  manner,  if  ^  (a+oc) 
be  multiplied  by  ^  (a+a;)^,  the  product  will  he  a-^x. 

289.  Hence,  if  the  numerator  and  denominator  of  a  fraction 
of  the  form  - —  be  multiplied  by  y  x^~^j  the  result  will  be  a 

'Y   X 

fraction,  whose  denominator  shall  be  rational. 

Thus,  let  both  the  numerator  and  denominator  of  the  frac- 
tion   be  multiplied  by  ^x,  and  it  becomes  ;  and  by 

^x  X 

multiplying  the  numerator  and  denominator  of  the  fraction 

,,y    .     ,,  by  y  (a-\-x)^,  it  becomes  yLlA-^-l-^  =     ^  /    '-, 
y{a-{-xy    ^^^         ''  y  {a^^f  a+a? 

Or,  in  general,  if  both  the  numerator  and  denominator  of  a 

fraction  of  the  form  ~ —  be  multiplied  by  y  a;"~^  it  becomes 
y  X 

a\/  a:" — ^ 

— ,  a  fraction  whose  denominator  is  a  rational  quan- 

X 

tity. 

290  Compound  surd  quantities  are  such  as  consist  of  two 
or  more  terms,  some  or  all  of  which  are  irrational ;  and  if  a 
quantity  of  this  kind  consist  only  of  two  terms,  it  is  called  a 
binomial  surd  ;  and  a  fraction  whose  denominator  is  a  binomial 

surd,  is,  in  general,  of  the  form  —. 

y  a±y  b 

291.  If  a  multiplier  be  required,  that  shall  render  any  bi- 
nomial surd,  whether  it  consist  of  even  or  oc?c?  roots,  rational,  it 
may  be  found  by  substituting  the  given  numbers,  or  letters,  of 


214  IRRATIONAL  QUANTITIES. 

which  it  is  composed,  in  the  places  of  their  equals,,  in  the  fol- 
lowing general  formula : 
Binomial,  y/  aJti\/  b. 

Multiplier,  y  a^-^^^  ^"""^*  +  V  «"~^*^  f  V  a^-^b^-{-,  &c., 
where  the  upper  sign  of  the  multiplier  must  be  taken  with  the 
upper  sign  of  the  binomial,  and  the  lower  with  the  lower  ; 
and  the  series  continued  to  n  terms.  This  multiplier  is  de- 
rived from  observing  the  quotient  which  arises  from  the  actual 
division  of  the  numerator  by  the  denominator  of  the  following 
'fractions :  thus, 

I.   ^■=x^—'^  +  x^—^y  +  x^—'^y'^  +,  (fee.  .  4-^*"^  to  ^ 

X — y 

terms,  whether  n  be  even  or  odd,  (Art.  108). 

a;" 1/" 

II..  ^-=a;"— ^— a;"— 2w  +  a:"— ^w2  — ,  &c.     .     .   — t/"— '  to 

x-\-y  y  y       ^  J 

n  terms,  when  »  is  an  even  number,  (Art.  109). 
x"  -4~  if* 
III.    — -^—a:"— 1— a;"— 2y-f-a;"— V— J  &c.      .     .      +v»— ^ 
x-\-y  :}  3  •  :/ 

to  n  terms,  when  n  is  an  odd  number,  (Art.  110). 

292.  Now  let  a;"=a,   y"=i  ;  then,    (Art.    116),    x^'^^  a, 

y=.y  h,    and    these   fractions    severally   become  ;^ — l, 

■^  a — y^  0 

— -,  and 7  ;  and  by  the  application  of  the  rules 

Ya-\-yb'      ya+yb'         ^        ^^ 

in  the  preceding  section  we  have  a;"— ^  =y  a"— ^ ;  a^— 2=^  a"— 2, 
a:"— 3— y  a»— 3,  &c.  also,  y'^=y  b"^  ;  y'^=y/  b^  \  &c.  ;  hence, 
y,n-2y—ny  an-^xy  b=Y  a^-H  ;  x'^-^y^^y  a^-^xy  b^=y  a 
n—^3  .  ^Q  gy  substituting  these  values  of  a;**— ',a;'»— ^y,  a;"— y, 

&c.,  in  the  several  quotients,  we  have  — —--=y  a—^-\- 

y  d — y  0 

y  a''—'^b-\-y  a«— 3Z>2_j-,  <fec -Vy  Z>"— 1  to  w  terms  ;  where 

n  may  be  any   whole  number  whatever.     And  — — \-m  = 
^  y  a-\-y  b 

y  gn—^—y  a^-2b-{.y  an-H"^  —  ,  &c.  .  . .  4zy  b"^^  to  n  terras  ; 

where  the  terms  b  and  y  5"  -^  have  the  sign  +,  when  n  is  an 

odd  number  :  and  the  sign  — ,  when  n  is  an  even  number. 

293.  Since  the  divisor  multiplied  by  the  quotient  gives  the 
dividend,  it  appears  from  the  foregoing  operations  that,  if  a 
binomial  surd  of  the  form  y  a—y  b  be  multiplied  by  y  a"— ^-f- 
y  a"— 2&  +  ,  &c.  .  -{-y  b^—'^  (n  being  any  whole  number  what- 
ever), the  product  will  be  a  — 6,  a  rational  quantity  ;  and  if  a 
binomial  surd  of  the  form^  a+y  b  be  multiplied  by  y  a"— ^ 
—y  a"— 26+5/  »— 3i2_^  (5^c.  .  .  .  ±y  b"^^,  the  product  will  be 


IRRATIONAL  QUANTITIES.  213 

a-\-h  or  a—b,  according  as  the  index  n  is  an  odd  or  an  even 
number. 

294.  Hence  it  follows,  that,  if  the  numerator  and  denomina- 
tor of  the  fraction  (Art.  290),  be  multiplied  by  the  multiplier, 
(Art.  291),  it  becomes  another  equivalent  fraction,  whose  deno- 
minator shall  be  rational. 

There  are  some  instances,  in  which  the  reduction  may  be 
performed  without  the  formal  application  of  the  rule,  which 
will  be  illustrated  in  the  following  examples. 

Ex.  1.  Reduce  ^^—7- — ^r-  to  a  fraction  with  a  rational 
yo  —  y  o 

denominator. 

To  find  the  multiplier  which  shall  make  -/5  — y^S  rational, 

we  have  n=z2,  a  =  5,  b  =  3  ;   .'.  (Art.  291),   y  a"-^-\-  y  a"-% 

=  (since  a"-2  =  a2-2=a'':zzl)    ^5  +  ^3;    .-.  :^0g±^X 

295.  This  multiplier,  ^^■\-  v/3,  could  be  readily  ascertain- 
ed, without  the  application  of  the  formula,  by  inspection  only  ; 
since  the  sum  into  the  difference  of  two  quantities  gives  the 
difference  of  their  squares  ;  also  the  multiplier  that  shall  render 
-y/a-f-'v/^  ''^cttional,  is  evidently  -y/a — -y/b.  In  like  manner,  a 
trinomial  surd  may  also  be  rendered  rational,  by  changing  the 
sign  of  one  of  its  terms  for  a  multiplier ;  and  a  quadrinomial 
surd  by  changing  the  signs  of  two  of  its  terms,  &c. 

2 

Ex.  2.   Reduce     .    . — — 7-  to  a  fraction  with  a  rational 

V5+^3  — -v/2 

denominator.  • 

In  the  first  place,     ,.       '1        ,    X^^^^^^^^S 
^  VS+VS  — V2     y5  +  '/3  +  V2 

2(1/54-1/34-^2).     .    •V/54-V34-V2      -34-Vl5_ 


64-2v'15  '  34-VI5  _34-'/15"~' 

(V5+/3+V2)2^3tVl5)  .^  ^^^  ,^^^,.^„  ^^^^.^^^ 

Ex.  3.  Reduce  ^ — - — 5 — -  to  a  fraction  with  a  rational  de- 
^3  —  ^2 

nominator. 

To  find  the  multiplier  which  shall  make  ^3  —  ^2  rational, 
we  have  w=r3,  anz3,  b=2  ;  .'.  V  a"-! 4-  y  a"-'^b-\-  U  5"-'  = 
3/94.3/6  +  3/4. 

Now  (3/3-V2)(V9  +  ^64-^4)  =  a-6=3-2  =  l; 


216  IRRATIONAL  QUANTITIES. 

.  the  denominator  is  1,  and  the  fraction  is  reduced  to  ?/  9-{- 

296.  Hence  for  the  sum,  or  difference,  of  two  cube  roots, 
which  is  one  of  the  most  useful  cases,  the  muhiplier  will  be 
a  trinomial  surd  consisting  of  the  squares  of  the  two  given 
terms,  and  their  product,  with  its  sign  changed. 

Ex.  4.  Reduce      V,  J  ■   to  a  fraction  with  a  rational 

denominator.  Ans. -—. 

2 
3 

Ex.  5.  Reduce  —r- y-  to  a  fraction  with  a  rational  de- 

V  ^  —  y  ^ 

nommator.  Ans.  —^ — 

5 — X 

o 

Ex.  6.  Reduce     .    , — .     ,  ^  to  a  fraction  whose  denomi 
V3+V2  +  1 

nator  shall  be  rational.  Ans.  4-f-2'y/2— 2'y/6. 

Ex.  7.  Reduce  ^- — T-h-r-  to  a  fraction  whose  denominator 
shall  be  rational. 

2 
Ex.  8.  Reduce  ^    ^  ,  4  ,  q  *^  ^  fraction  whose  denominator 

shall  be  rational.  Ans.  ^125  —  '*/  75-f  ^  45-^  27. 

297.  It  may  not  be  improper  to  take  notice  here  of  another 
transformation  which  binomial  surd  quantities  may  undergo 
by  equal  involution,  and  evolution. 

Ex.  1.  To  transform  -v/24-'y/3  to  a  universal  surd. 

Its  square  =5  +  2 -/e  ;  .-.the  root  = -/(5-h2-/6). 

Ex.  2.  To  reduce  ^21 -\-  -v/48  to  a  universal  surd. 

Here  (  v'274-'v/48)2=27+2V'l296+48=147  ;  .'.-^21 
4.'/48  =  'v/147=  V49  X  3=:7-v/3. 

Ex-  3.  To  transform  ^  320  —  ^  40  to  a  general  surd. 

Here  (^  320-3/  40)3^320-33/  4096000  +  3  ^  512000 
—40=40;  .-.^320-^40=2^5. 

298.  This  transformation  is  very  useful,  since,  by  means 
of  it,  we  can  always  reduce  the  sum  or  difference  of  any  two 
surd  quantities,  if  they  admit  of  the  same  irrational  part,  to  a 
single  surd.  This  may  be  proved,  in  general,  thus  ;  if  y/  a  and 
y/  b  admit  of  the  same  irrational  part,  they  must  be  of  the 
form  7  a'"m  and  y  b'"m  ;  and  (  7  a"'wi+  y  b"'mY=za"'m+n 


IRRATIONAL  QUANTITIES.  217 

b"'m=a"'m-^na"^^'Xmb"'-^&c b'^'m  .-.y  a-\-y  b=y 

(a'"m-{-nma'"—'^y'*-\-&,c i^''m)=  the  nth  root  of  a  rational 

quantity.  Hence  the  product  of  -y/a  by  -x/b  is  rational  if 
-y/a  and  -y/b  admit  of  the  same  irrational  part;  also,  ^  a^X 
y  b,  or  y  axy  b'^,  is  rational,  if  ^  a  and  y  b  admit  of  the 
same  irrational  part ;  and,  in  general,  y  a  -'^  xy  6,  or  ^  c  a 
y  5*^^,  is  rational,  if  y  a  and  y  b  admit  of  the  same  irrational 
part. 

299.  It  is  propel  to  observe,  that,  for  the  addition  or  sub- 
traction of  two  quadratic  surds,  the  following  method  is  given 
in  the  Bija  Ganita,  or  the  Algebra  of  the  Hindoos,  translated 
by  Strachey.  Thus,  to  find  the  sum  or  difference  of  two  surds, 
^a  and  -y/b,  for  instance. 

RULE. 

Call  a-^b  the  greater  surd  ;  and,  if  aX&  is  rational,  (that 
is,  a  square)^  call  2  ^/ab  the  less  surd,  the  sum  will  be  '\/(a-{-b 
■\-2^/ah)^  (nr  ( V^iV^)^)'  ^"^^  ^^®  difference  '\/{a-\-b — 
2'v/a5).  If  a  X  &  is  irrational,  the  addition  and  subtraction  are 
impossible  ;  that  is,  they  can  only  be  indicated. 

Example.  Required  the  sum  and  difference  of  -y/2  and  -y/S. 
Here  2  +  8  =  10=>  surd;  2  x8z=16, .-. 'v/16=:4,  and2Vl6 
=2x4  =  8=<surd.  Then  10  +  8  =  18,  and  10—8  =  2; 
'.  '/18=  sum,  and  -^2=  difference. 

another  rule. 

Divide  a  by  J,  and  write  /^  in  two  places.  In  the  first 
place  add  1,  and  in  the  second  subtract  1  ;  then  we  shall  have 
7^(71+ 0'^*J=  V«+V*,  and^[(7^-l)^  X  J]  =  -/a 

irrational,  (that  is,  not  a  square),  the  addition  or 

subtraction  can  be  only  made  by  connecting  the  surds  by  the 
signs  4-  or  —  1,  as  they  are. 

Sturmius,  in  his  Mathesis  Enucleata,  has  also  given  a  me- 
thod similar  to  the  above. 

Ex.  4.  To  transform  -/  2+-y/3  to  a  general  surd. 

Ans. /(5+2/6). 
20 


'^Jv^ 


218  IRRATIONAL  QUANTITIES. 

Ex.  5.  To  transform  ^/a—1^/x  to  a  universal  surd. 

Ans.  -/(o+^a:— 4^ar). 
Ex.  6.  To  transform  3^/  \^^-^/  72  to  a  universal  surd. 

Ans.  33/  9. 

§  V.    METHOD  OF  EXTRACTING  THE  SQUARE  ROOT  OF  BINOMIAL 
SURDS. 

300.  The  square  root  of  a  quantity  cannot  be  partly  rational  and 
partly  a  quadratic  surd.  If  possible,  let'  -/n=a4-  -y/m  ;  then  by 
squaring  both  sides,  ni=a2-l-2a'\/m-|-OT,  and  2a-v/m=n — a^ — m\ 

fl  __  q2 jj^ 

therefore,  -^m^z ,  a  rational  quantity,  which  is  con- 

trary  to  the  supposition. 

A  quantity  of  the  form  ■\/a  is  called  a  quadratic  surd. 

301.  If  any  equation  x-\-  -^yzzza-^-^b,  consisting  of  rationa* 
quantities  and  quadratic  surds,  the  rational  parts  on  each  side  are 
equal,  and  also  the  irrational  parts. 

If  X  be  not  equal  to  a,  let  x=.a  +  m  ;  then  a-\-m-\--\/y=a 
-\-^/b,  or  m+ Vy=  "v/i  ;  that  is,  -y/b  is  partly  rational,  and 
partly  a  quadratic  surd,  which  is  impossible,  (Art.  300) ;  .'.x=za, 
andy'y=^i. 

302.  If  two  quadratic  surds  -y/x  and  y/y,  cannot  be  reduced  to 
others  which  have  the  same  irrational  part,  their  product  is  irra- 
tional. 

If  possible,  let  ■\/xy=rx,  where  r  is  a  whole  number  or  a 
fraction.  Then  xyz=r^x'^,  and  y=r^x  ;  .'.  \/y=r-\/x  ;  that  is, 
•y/y  and  -\/x  may  be  so  reduced  as  to  have  the  same  irrational 
part,  which  is  contrary  to  the  supposition. 

303.  One  quadratic  surd,  -y/x,  cannot  be  made  up  of  two  others, 
•yjm  and  -yjn,  which  have  not  the  same  irrational  part. 

If  possible,   let  'y/x=:i-\/m-\-^  n  \  then  by  squaring  both 
sides,   x=zm-\-2^  mn-\-n,  and  x — m—n-=z2y^  mn,  a  rational 
quantity  equal  to  an  irrational,  which  is  absurd. 
1^ 

304.  Let  {a-\-b)c  =ix-{-y,  where  c  is  an  even  number,  a  a  ra- 
tional quantity,  b  a  quadratic  surd,  x  and  y,  one  or  both  of  them, 

1 
quadratic  surds,  then  {a  —  6)c=a: — y. 

c 1 

By  involution,  a+b=x'+cx''-^y{'C.—-—  x'-^y^-^-  &c.,  and 

since  c  is  even,  the  odd  terms  of  the  series  are  rational,  and 

c— 1 
the  even  terms  irrational ;  .•.a=x'-\-c.  ^——af-^y^  +  &c.,  and 

2 

i=c^-iy-fc.^.^-^a;^-3y34-  &c.,  (Art.  301);  hence,  a-b 
2        o 


IRRATIONAL  QUANTITIES.  219 

c— 1 

=xc—cxc—'^i/-{-c  .  — - — x'^—'^xp' — ,  &c. ;  and  consequently,  by 
tit 
1 
evolution,  {a  —  hy  z=x—y. 

305.  If  c  he  an  odd  number,  a  and  b,  one  or  both  quadratic 
surds,  and  x  and  y  involve  the  same  surds  that  a  and  h  do  re- 

i_  i_ 

spectivehj,  and  also  (a-\-.b)<=  =x-\-i/,  then  {a—h)'^=x—y. 

Q 1 

By  involution,  a-{-hz=zx'^-\-cx^~'^y-{-c  .  —;^x<^~^y^-^,  &,c., 

where  the  odd  terms  involve»the  same  surd  that  x  does,  be- 
cause c  is  an  odd  number,  and  the  even  terms,  the  same  surd 
that  y  does  ;  and  since  no  part  of  a  can  consist  of  y  and  its 

c 1 

parts,  (Art.  301),  a=x'-\-c  .  —p—x<^—'^y'^-\-,  &c.,and5=:ca;c— ^y 

-\-c  .  — - —  .  — —  .  x^—'^y'^-\-,  &c. ;  hence,  a—h^zx^—cx'^—'^y 

til  o 

c— 1  L 

+  c  .  a;c— y~-,  &LC.  ;  .-.  by  evolution,  (a— &)<^=a;— y. 

4/ 

306.  The  square  root  of  a  binomial,  one  of  whose  terms  is  a 
quadratic  surd,  and  the  other  rational,  may  sometimes  be  ex- 
pressed by  a  binomial,  one  or  both  of  whose  terms  are  quadratic 
surds. 

Let  a-\'-\/b  be  the  given  binomial,  and  suppose  ■\/{a-\--}/b) 
=zx-\-y  ;  where  x  and  y  are  one  or  both  quadratic  surds  ;  then 
•\/(a—-y/b)z=x—y',  .'.by  multiplication,  ■}/{a'^ — b)^=x^—y^, 
also,  by  squaring  both  sides  of  the  first  equation, 
a-^^b=:x^-\-2xy-\-y'^,  and  a=a;2-j-y2. 

.-.  by  addition,  a-{-^(a'^  —  b)=2x^,  and  by  subtraction,  a— 
-v/(a2— 5)=2y2  ;  and  the  root  x-\-y=  ^aa-{-^^(a'^—b)]  + 
V[ia--W{a'-b)]. 

From  this  conclusion  it  appears,  that  the  square  root  of 
a+\/b  can  only  be  expressed  by  a  binomial  of  the  form  x-^y, 
one  or  both  of  which  are  quadratic  surds,  when  a'^—b  is  a  per- 
fect square. 

By  a  similar  process  it  might  by  shown  that  the  square  root 

of  a-yb  is  V[^a  +  iV(«^-^)]- VB^-iVCa^-^*)],  sub- 
ject to  the  same  limitation. 

Ex.  1.  Required  the  square  root  of  3  +  2-^2. 

Let  -v/(3+2-v/2)=a:+y;  then  ^(3—2y2)  =  x-~y;  by 
multiplication,  ■^{9  —  8)=:x^—y^;  that  is,  x^—y^=l. 

Also,  by  squaring  both  sides  of  the  first  equation,  3+2'v/2 
=x^-{-2xy—y^,  dind  x'^-jry^=3;  /.by  addition,  2a;2:=4,  and 
x=^2. 


220  IRRATIONAL  QUANTITIES. 

Again,  by  subtraction,  2y2— 2  ;  .-.^=1,  anda;4-y=V2  +  l 
=  the  root  required. 

Or,  the  root  may  be  found  by  substituting  3  for  a,  2-^2  = 
-y/S  for  -y/h,  or  8  for  Z>,  in  the  above  formula  ;  thus, 
a:+y= V[f +iV(9-8)]  + -/B  -  i  V(9-8)  =  ^(1+^)+ 

£x.  2.  Required  the  square  root  of  IQ+S^/S. 

Ans.  4+V3 
Ex.  3.  What  is  the  square  root  of  12  — -v/UO  ? 

Ans.  -/?— -/S 
Ex.  4.  Find  the  square  root'of  7+4^3. 

Ans.  2  +  V3 
Ex.  5.  Find  the  square  root  of  7— 2^/10. 

Ans.  -v/5  — -/2 
Ex.  6.  Find  the  square  root  of  31  +  12-y/— 5. 

Ans.  64--;/"^ 
Ex.  7.  Find  the  square  root  of  18  — 10  v'— 7. 

Ans.  5  — -y/— 7 
Ex.  8.  Find  the  square  root  of  —1+4-/— 5. 

Ans.  2  +  V— 5 
307.   The  cth  root  of  a  hinomial^  one  or  both  of  whose  terms 
tre  possible  quadratic  surds,  may  sometimes  be  expressed  by  a 
binomial  of  that  description. 

Let  A+B  be  the  given  binomial  surd,  in  which  both  terms 
are  possible  ;  the  quantities  under  the  radical  signs  whole 
numbers  ;  and  A  is  greater  than  B. 

Lety  [(A-i-B)xVQ]=^+y; 

then    ^[(A-B)x-v/Q]=a;-y; 

.-.by  multiplication,  ./ [(A2- B2)xQ]=ar2—y2.    ^q^  \^^ 

Q  be  so  assumed,  that  (A^— B^jxQ  may  be  a  perfect  cth 
power  =«",  then  x^ — y'^^=.n. 

Again,  by  squaring  both  sides  of  the  first  two  equations,  we 
have 

^  [(A+B)2xQ]=a;2-h2a:y+y2 
^  [( Ar-B)2  X  Q]  ==a:2_2a:y+y2  . 

.-.  ^  [  (A  +  B)2xQ]+{/  [(A-B)2xQ]=2a:24.2y2;  which  is 
always  a  whole  number  when  the  root  is  a  binomial  surd  ;  take 
therefore  s  and  i,  the  nearest  integer  values  of{/  [(A+B)2x 
Q]  andy  [(A  — B)2xQ],  one  of  which  is  greater  and  the 
other  less  than  the  true  value  of  the  corresponding  quantity ; 
then  since  the  sum  of  these  surds  is  an  integer,  the  fractional 
parts  must  destroy  each  other,  and  2x^-\-'Zy'^=is-\-t,  exactly, 
when  the  root  of  the  proposed  quantity  can  be  obtained.  We 
have  therefore  these  two  equations,  x^—y'^z=.n,  and  as^-f  y2__^^ 


IMAGINARY  QUANTITIES.  221 

-fif ;  .'.  by  addition,  2x^=n-\-^s-]-lt,  and  x=^y(2n-{-s-^t) ; 
and  by  subtraction,  2i/'^=^s-\-^t  —  n,  and  i/=^y/{s-^t—2n). 

Consequently,  if  the  root  of  the  binomial  -y^  [(A+B)  X  VQ>] 
be  of  the  forma;+y,  it  is  ^■\/(2n-i-s-\-t)-{-^-\/(s-\-t—-2n) ;  and 

the  cih  root  of  A+B  is  -5-^ ■ —    '  'If^ -'. 

^  \/  Q. 

Ex.   1.  Required  the  cube  root  of  10  +  -/ 108. 

In  this  case,  ^108  is  >10;  .-.  A=  ^108,  B  =  10,  A2— B2 
=  108— 100  =  8,  and  8Q  =  n^.  Now,  since  8  is  a  cube  number, 
Q  may  be  taken  equal  to  1  ;  then8Q  =  8=n^;  .-.  n=2.  Also, 
3/[(A+B)2]=7+/;  ^[(A-B)2]  =  l-/,  where /is  some 

fraction  less  than  unity  ;  .-.  ^=7,  t=l  ;  and  a7+y=-^^^ — 

-V3  +  1. 

If  therefore  the  cube  of  10+ -y/ 108  can  be  expressed  in  the 
proposed  form,  it  is  -y/S+l  ;  which  on  trial  is  found  to  suc- 
ceed. 

Ex.  2.  Find  the  cube  root  of  26+ 15  ^3. 

Ans.  2  +  -v/3 
Ex.  3.  Find  the  cube  root  of  9 -/ 3  —  1 1^2- 

Ans.  '/3  — 'v/2. 
Ex.  4.  Find  the  cube  root  of  4^5 +8. 

A         V5+1 

Ans.      „^^    . 

308.  In  the  operation,  it  is  required  to  find  a  number  Q,  such 
that  (A^ — B)2xQ  may  be  a  perfect  cth  power  ;  this  will  be 
the  case,  if  Q  be  taken  equal  to  (A^ — l^^y-'^ ;  but  to  find 
a  less  number  which  will  answer  this  condition,  let  A^  — B^  be 
divisible  by  a,  a,  ...  (m) ;  6,  6,  .  .  .  (n) ;  d,  d,  ...  (r) ;  &c. 
in  succession,  that  is,  let  A'^-^B'^z=a'"b"d'',  &c  ;  also,  let  Q  = 
a'b^d^  &c.  Then  {A'2^B^).Q  =  ar*^  xb"*^  X  d"*',  &c.  which 
is  a  perfect  cth  power,  if  x,i/,  z,  &c.,  be  so  assumed  that  m+x, 
w+y?  ^+^>  <fec.  are  respectively  equal  to  c,  or  some  multiple 
of  c.  Thus,  to  find  a  number  which  multiplied  by  2250  will 
produce  a  perfect  cube,  divide  2250  as  often  as  possible  by 
the  prime  numbers  2,  3,  5,  &c.  and  it  appears  that  2x3x3 
X5x5x5=2x32x  53=^2250  ;  if,  therefore,  it  be  multiplied 
by  22  X  3,  it  becomes  2^  x  3^  x  5^,  or  (2.3.5)3  .  a  perfect  cube. 
See  Wood's  Algebra 

§  VI.    CALCULATION  OF  IMAGINARY  QUANTITIES. 

309.  In  the  Involution  of  negative  quantities,  it  was  ob- 
served, that  the  even  powers  were  all  aflTected  with  the  sign  +, 
and  the  odd  powers,  with  —  ;  there  is  consequently  no  quan 

20* 


222  IMAGINARY  QUANTITIES. 

tity  which,  multiplied  into  itself  in  such  a  manner  that  the  num- 
ber of  factors  shall  be  even,  can  generate  a  negative  quantity. 
Hence  quantities  of  the  form  -y/ — a^,  y  — 16,  |/  — ci^,y/  —a*, 
and  in  general  y/  —  a,  have  no  real  roots  ;  and  are  therefore 
usually  called  impossible  or  imaginary. 

It  is  to  be  observed  that  all  quantities,  either  positive  or  ne- 
gative,  or  even  irrational,  are  considered  to  be  real. 

310.  Although  the  values  of  imaginary  quantities  are  un- 
assignable in  numbers,  they  are  yet  of  great  use  in  some  of 
the  higher  branches  of  analysis,  as  well  as  in  showing  when 
a  result  of  this  kind  occurs,  that  the  question,  under  the  pro- 
posed conditions,  is  impossible. 

Thus,  if  it  should  be  required  to  find  a  number  whose  square 
subtracted  from  3,  gives  7  for  a  remainder.  We  have  for  a 
translation 

^-x^=.l',  .-.  a;2=3-7  =  — 4. 

The  unknown  quantity  x  is  therefore  the  square  root  of  the 
number  — 4,  a  root  which  is  imaginary  ;  and  in  fact,  the 
enunciation  comprehends  an  impossibility.  If  we  had  thus 
proposed  the  question,  to  find  a  number  whose  square  added  to 
3,  gives  7  for  a  sum,  we  should  have  had  for  the  translation 
a;24-3=7  ;  .-.  x'^=.^  and  07=2,  which  is  a  real  root. 

Thus  negative  isolated  results  arise  from  the  subtraction  of 
a  greater  number  from  a  lesser,  and  imaginary  quantities  are 
given  by  a*new  operation  to  be  performed  upon  these  kind  of 
remainders. 

311.  This  being- premised,  it  is  only  necessary  farther  to 
observe,  that  the  method  of  adding  and  subtracting  imaginary 
radicals,  is  the  same  as  for  real  quantities. 

Thus,  y -a+2y -a:=:3y -rt  ;  6+ V-4  +  6--v/-4 
=  12;  and  "i  y/ —ax-\-^  —y—{^J —ax^^  —y)—2-^ —ax 
+2{/  -y. 

312.  Every  imaginary  radical  quantity  of  the  form  •/ — a, 

can  be  reduced  to  the  form  y/a  X  -\/  —  1,  or  a/^-y/  —1. 

In  order  to  demonstrate  this,  let  the  identical  equality  be, 
(c—b)a:=(c—,b)a',  by  extracting  the  root  of  both  sides,  we 
shall  have  ■y/{c—b)x-y/a=y/[(c  —  b)a]',  which  under  the 
relation  &>c,  or  in  the  hypotheses,  for  instance,  b=c-^\,  be- 
comes -/  — IX  V'^='/— «  ;  and,  in  general,  2^  —azzz'^y  aX 

It  may  be  demonstrated,  in  a  similar  manner,  that     j 


IMAGINARY  QUANTITIES.  223 

313.  Hence,  in  the  calculation  of  imaginary  radicals,  it  is 
sufficient  to  demonstrate  the  rules  for  multiplying  and  involving 
the  imaginary  radical  -y/  — 1  ;  since  imaginary  quantities  can 
be  always  resolved  into  factors  ;  so  that  — 1  only  shall  remain 
under  the  radical  sign. 

314.  In  the  first  place,  then  it  may  be  observed,  when  a^ 
is  considered  abstractedly,  or  without  any  regard  to  its  gene- 
ration, then  -y/a^  maybe  either  -{-a  or  — a  there  being  no- 
thing in  the  nature  of  the  quantity  so  taken,  to  denote  from 
which  of  these  two  expressions  it  was  derived. 

315.  But  this  ambiguity,  which,  in  the  above  mentioned  case, 
arises  from  our  being  unacquainted  with  the  origin  of  the 
quantity  whose  root  is  to  be  extracted,  will  not  take  place  when 
the  sign  of  the  quantity  from  which  it  was  produced  is  known  ; 
as  there  can,  then,  be  only  one  root,  which  must  evidently  be 
taken  in  plus  or  minus,  according  to  the  state  it  existed  in  be- 
fore it  was  involved. 

316.  Thus,  V[(  +  «)  X(4],  or  -/[C  +  a^)]  cannot  be  of  the 
ambiguous  form  Jka,  as  it  would  have  been  if  a^  had  been  un- 
conditionally assumed,  but  it  is  simply  a  ;  and,  for  a  like  rea- 
son, V[(  —  a)x{  —  a)],  or  ^(  —  aY  is  =  —a,  and  not  J-a  ; 
since  the  value  of  the  equivalent  expression -(--v/^^'  ^^  — V^^ 
in  these  cases,  is  determined,  from  the  circumstance  of  its  be- 
ing known  how  a^  is  derived. 

317.  Hence  the  product  o/  -y/—  1  by  -y/— 1,  or  which  is  the 
same,  (■\/  —  l)2is  =  —  -\/l=z  —  1.  This  is  what  appears,  evi- 
dent from,  since  that  in  squaring  a  quantity  with  the  radical 
sign  ■;/,  we  have  only  to  take  it  away,  that  is,  to  pass  the 
quantity  from  under  the  radical  sign, 

318.  Also,  if  the  factors,  in  this  case,  be  both  negative,  the 
result  will  be  the  same  as  before  ;  since  — (-y/  —  l)x  — (-\/  — 1) 
=:+(\/  —  1)^=— 1  ;  but  if  one  of  the  factors  be  positive  and 
the  other  negative,  we  shall  have  +(V  —  1)X — (V  —  1)  =  — 
(V-l)'^=  +  l. 

319.  All  whole  positive  numbers  are  comprised  in  one  of  these 
fourformul(Bi 

4»,  4n-M,  4n+2,  4ra+3, 
n  being  a  whole  positive  number ;  since  that,  if  any  whole  num- 
ber be  divided  by  4,  the  remainder  must  be  0,  1,  2,  or  3. 


224  IMAGINARY  QUANTITIES. 

If  we  designate  y'  — 1  by  ar,  the  several  powers  of-/  — 1 
shall  be  therefore  represented  by  one  of  these  four  formulae  : 

(^_l)4»+i=:a;*"+i=:a;*".ir=a;=  +  V  — 1  ; 

Thus,  in  order  to  know  any  given  power  of  '^  —  I,  it  is sujp.- 
cient  to  divide  the  exponent  of  the  power  proposed  by  4,  anct  the 
power  of  ^/  —  \  indicated  by  the  remainder  shall  be  that  which 
is  required. 

320.  When  one  imaginary  quantity  is  to  be  multiplied  by  an- 
other, the  result  whether  they  be  both  positive  or  both  negative, 
is  equal  to  minus  the  square  root  of  the  product,  taking  them  as 
real  quantities. 

Thus,  (-f/ — a)x(4-/ —  ^)  =  — /«&  ;  since,  (  +  / — a) 
X(+^/  ~b)=^  ax^/  -IX/ix/  — l=/aX/6X 
(-/  —  1  )2  —  —  1  X ./  a5  =  — /  a5.  And,  in  a  similar  manner,  it 
may  be  proved  that  ( — /  —a)  X  ( — /  —b)—  —^/  ab. 

321 .  And  if  one  of  the  imaginary  radicals  be  positive,  and  the 
other  negative,  the  result  arising  fromtheir  multiplication  will 
be  plus  the  square  root  of  their  product,  taking  them  as  before. 

Thus,  {^\-^/—a)x(  —  ^—b  =  -{-'^ab•,    since   +  ^— a= 

+  y/aX  y/  —  l,  and  —^ — 5  =  (-.y  —  ljx -/*  ;    -'-{V^X 

y'_l)X(-^-l)xV*)=[(  +  V-l).(-V-l)]V'«6  = 
+  1  x-}/ab  =  +  -\/ab. 

322.  When  one  imaginary  radical  is  to  be  divided  by  another, 
the  result,  whether  they  be  both  positive  or  both  negative,  will  be 
equal  to  plus  the  square  root  of  their  quotient,  taking  them  as 
real  quantities. 

rri         +  V  —  <2         —  y/ —a  a  -f  -/ —  a 


323.  And  if  one  of  the  imaginary  radicals  be  positive  and  the 
other  negative,  the  result  arising  from  division,  will  be  minus 
the  square  root  of  their  quotient,  taking  them  as  before. 

Th"«.  Z^-i  0'  +7—4=  -  Vl ;  and  -^—  or 

--/-«=     1. 

+  -/-« 

324.  If  an  imaginary  radical  is  to  be  divided  by  a  real  radi- 
cal, or  a  real  radical  by  an  imaginary  one,  the  result  will  be  equal 
to  plus  or  minus  the  square  root  of  their  quotient,  according  as 
the  radical  is  affirmative  or  negative. 


IMAGINARY  QUANTITIES.  225 

Thus,  -i^-T-  or  -y—T=  +  /  — T '  ^^^  ^—7-  or  -7^=^^= 

The  several  powers  of  imaginary  radicals  can  be  readily 
derived  from  the  formulae  (Art.  319) ;  it  only  now  remains  to 
illustrate  the  preening  rules  by  a  few  practical  examples. 

Ex.  1.  It  is  required  to  multiply  a—^  —hhy  a—-/  —5,  or 
to  find  the  square  of  a — ■/  — b.  '        ' 

a-^-b 


dp- — a^ —  h 
—a\/  —  b—h 

a^—2a{^/~-b)—h  Ans 
Ex.  2.  It  is  required  to  find  the-  quotient  of  1  +  -y/""  ^  divid- 
ed by  1  — y  — 1. 

Ans. 

Ex.  3.  It  is  required  to  multiply  l  +  y'  — 1  by  l  +  y'— 1  ; 
or  to  find  the  square  of  \■\•^/  —  \.  Ans.  2-\/  —  1. 

Ex.  4.  It  is  required  to  find  the  product  arising  from  mul- 
tiplying 1 +-/— 1  by  1  —  V  —  1  Ans.  2. 

Ex.  5.  It  is  required  to  find  the  square,  or  second  power  of 
a^\■^/—W■.  Ans.  a2_^2_j.2a6^_l. 

Ex.  6.  It  is  required  to  multiply  5+2^—3  by  2  —  -/— 3. 

Ans.  16—/  —3. 

Ex.  7.  It  is  required  to  find  the  cube,  or  third  power,  ol 
a-/  ~y^.  Ans,  o?~'^a\P'^{ly^~^d^b)y/  — L 

Ex.  8.  It  is  required  to  find  the  quotient  of  3+y'  —4  di 
vided  by  3-2/  —1.  Ans.  Jg.(5+12/  —1) 

Ex.  9.  It  is  required  to  find  the  square  of  /  (a-j-J/  —1)4- 
/  (a— V  —1)-  Ans.  2a+2/  {a^^V^) 


226 


CHAPTER  VIII. 


ON 

PURE  EQUATIONS. 

325.  Equations  are  considered  as  of  two  kinds,  called  sim- 
ple or  pure,  and  adfected ;  each  of  which  are  differently  de- 
nominated according  to  the  dimensions  of  the  unknown  quan- 
tity. 

326.  If  the  equation,  when  cleared  of  fractions  and  radical 
signs  or  fractional  exponents,  contain  only  the  first  power  of  the 
unknown  quantity,  it  is  called  a  simple  equation. 

327.  If  the  unknown  quantity  rises  to  the  second  power  or 
square,  it  is  called  a  quadratic  equation. 

328.  If  the  unknown  quantity  rises  to  the  third  power  or  cube, 
it  is  called  a  cubic  equation,  &lc. 

329.  Pure  equations,  in  general,  are  those  wherein  only  one 
complete  power  of  the  unknown  quantity  is  concerned.  These  are 
called  pure  equations  of  the  first  degree,  pure  quadratics,  pure 
cubics,pure  biquadratics,  Sic,  according  to  the  dimension  of  the 
unknown  quantity. 

Thus,  x  —  a  +  b  is  a.  pure  equation  of  the  first  degree; 
cc^=:a^-{-ab  is  a.  pure  quadratic  ; 
x^=:a^-^a'^b-\-c  is  a  pure  cubic  ; 
x'^z=a^-\-a^b-{-ac'^-\-d  is  3, pure  biquadratic  ;  &c. 

330.  Adfected  equations  are  those  wherein  different  powers  of 
the  unknown  quantity  are  cohcerned,  or  are  found  in  the  same 
equation.  These  are  called  adfected  quadratics,  adfected  cubics, 
adfected  biquadratics,  &c.,  according  to  the  highest  dimension 
or  power  of  the  unknown  quantity. 

Thus,  x'^-\-ax  =  b,  is  an  adfected  quadratic 
x'^-\-ax^-{-bx=c,  an  adfected  cubic  ; 
x'^-^ax^-^bx'^-{-cx=:d,  an  adfected  biquadratic. 
In  like  manner  other  adfected  equations  are  denominated  ac- 
cording to  the  highest  power  of  the  unknown  quantities. 

§   I.    SOLUTION  OF  PURE  EQUATIONS  OF  THE  FIRST  DEGREE 
BY  INVOLUTION. 

331.  We  have  already  delivered,  under  the  denomination  of 
Simple  Equations,  the  methods  of  lesolviiig  pure  equations  of  the 


PURE  EQUATIONS.  227 

first  degree,  in  all  cases,  fexcept  when  the  quantity  is  affected 
with  radical  signs  or  fractional  exponents,  in  which  case  the 
following  rule  is  to  be  observed. 


RULE. 

332.  If  the  equation  contains  a  single  radical  quantity^ 
transpose  all  the  other  terms  to  the  contrary  side  ;  then  in- 
volve each  side  into  the  power  denominated  by  the  index  of 
the  surd ;  from  whence  an  equation  will  arise  free  from  radi- 
cal quantities,  which  may  be  resolved  by  the  rules  pointed  out 
in  Chap.  III. 

If  there  are  more  than  one  radical  sign  over  the  quantity, 
the  operation  must  be  repeated  ;  and  if  there  are  more  than 
one  surd  quantity  in  the  equation,  let  the  most  complex  of 
those  surds  be  brought  by  itself  on  one  side,  and  then  proceed 
as  before. 

Ex.  1.  Given  -^  (4a?4-16)=12,  to  find  the  value  of  a;. 
Squaring  both  sides  of  the  equation,  4a;+16=:144  ; 

by  transposition,  407=144  —  16  ;  .-.  a;=32. 
Ex.  2.  Given  y  (2a;+3)  +  4=::7,  to  find  the  value  of  x. 

By  transposition,  ^  (2a:-|-3)rr7— 4=3  ; 
cubing  both  sides,  2x-f  3=27; 
by  transposition,  2a:=27  — 3  ;  .-.  a;=12. 
Ex.  3.  Given  -/(12+j;)  =  2-f -/a:,  to  find  the  value  of  x. 
Bysquaring,  124-a:=4-}-4'/a;+a;; 
by  transposition,  8=4  ^Jx^  or  \/x=2  \ 
.'.  by  squaring,  «=4. 
Ex.  4.  Given  V(^+ 40)  =  10  —  ^0:,  to  find  the  value  of  x. 
By  squaring,  x-\-A:0=:l00—20^/x-\-x  ; 
by  transposition,  20-y'a;=60,  or  -v/a:=3  ; 
.-.by  squaring,  a;=9. 
Ex.  5.  Given  -/(a:— 16)=8  — -/ar,  to  find  the  value  of  a:. 
By  squaring  both  sides  of  the  equation, 

a;— 16  =  64  — 16-/^:+^  ;  .'.  16ya:=64  +  16  =  80  ; 
by  division,  ^/x=b  ;  .*.  a;=25. 
Ex,  6.  Given  ^(x—a)—  y/x—^  -y/a,  to  find  the  value  of  x. 
Squaring  both  sides  of  the  equation, 

x  —  a=x — '\/{ax)-^^a; 

.-.by  transposition,  ^(ax)  =  ^a; 

25a2  25a 

by  squaring,  ax=-j^ ;    .'.  ^^-^q- 

Ex.  7.  Given  -y/SX 'y/(a;+2)  =  -/5a;+2,  to  find  the  value 
of  a?. 


238  PURE  EQUATIONS. 

By  squaring,  5a;+10=:5a;4-4'v/5a;+4  ; 

by  transposition,  6  =  4  v^5a? ;  .-.  ^5x=^  ; 

by  squaring  again,  5x=z^  ;  .-.  x=^. 

Ex.  8.  Given  — 7 — =-^,  to  find  the  value  of  a;. 

yx         X 

Multiplying  both  sides  of  the  equation  by  -/a;, 

X  1 

X — ax=z-=l,  or  (l—a)x^=l  ;  .-.a-rr- . 

a?  1 — a 

T^      «    ^-         V^4-28     -v/x+38        ^    ,  ,,         ,         c 

Ex.  9.  Given  — — ■ =-^ ,  to  find  the  value  of  x. 

Va:+4        yx-f-6 

Multiplying  both  sides  by  ( ^/x•\•4:)  x  (  ^x-\-6), 

we  have  x-[-34:yx-{-l68=x+42^x-^152  ; 

by  transposition,  I6=z8^x,  or  2  =  -/ a?; 

.-.  by  squaring,  a:=r4. 

T,      ,^    ^.         \/ax—b     3^ax—2b  ,    ^    , ,,         ,        ^ 

Ex.  10.  Given  ~^, --r:z=~-. -7,  to  find  the  value  of  x. 

yax-\-o     Syax-j-DO 

Multiplying  both  sides  by  (-/aa;+6)x(3-/aa;-f-56), 

Sax +21  y/ax—5b^  —  3ax  +  b-)/ax~2b^y 

.-.by  transposition,  by^axz=3b^  ; 

by  division,  ^ax  =  3b; 

*.•  by  squaring,  ax=9b'^f  and  x= —  . 

Ex.  11.  Given -/(a;4- V^?)  — -/(a?— 'v/^)= 

I    /  ( 7-r  l»  to  find  the  value  of  x. 

Multiply  both  sides  of  the  equation  by  y^{x+  ^x),  x+ 

v'(«)-^(x^-«)=i|^, 

.'.  by  transposition,  x ^r— =  \^{x^~x) ; 

and  dividing  by  -y/a:,  -/a:— J=  -/(a:— 1)  ; 

.♦.by  squaring,  x—'^x-{-^=x—l  ;  .-.  -y/aj^-J, 

25 
and  by  squaring,  aj^rT^. 

Ex.  12.  Given  /  (a;— 24)  =  -v/a;— 2,  to  find  the  value  of  x. 

Ans.  a?=49. 
Ex.  13.  Given  ^  (4a 4- a:) =2/  {b+x)  —  \/x,  to  find  the  va- 

lue  of  X.  Ans.  x='-^^---^. 

Ex.  14.  Given  x+a-\-y/  (2ax+x^)  =  bf  to  find  the  value  of  a;. 


PURE  EQUATIONS.  229. 

Ex.  15.   Given  ^ ^  =  —. — 7-777,  to  find  the  value  of  a;. 


Ans 


-=(^)' 


Ex.  16.   Given—— — —-=14-^—— ,  to  find  the  value 

y^dx-|-l  2 

of  X,  Ans.  a:  =  3. 

Ex.  17.  Given  x=y[a^+x\^(b^+x^)]—a,  to  find  the  va- 

lue  ol  X,  Ans.  x= — . 

4a 
4 

Ex.  18.    Given    y/{2+x)+'i/x=-j- -. ,  to  find  the  va- 

y{2-\-x) 

lue  of  a;.  Ans.  x=-. 

Ex.  19.  Given  ^  (10a;+35)  — 1=4,  to  find  the  value  of  x. 

Ans.  a;=:9. 
Ex.  20.  Given  ^  (9x—4)  +  6=8,  to  find  the  value  of  x. 

Ans.  x=4, 
Ex.  21.  Given  y/{x-\-i6)=z2-{-^^,  to  find  the  value  of  x. 

Ans.  a;=9. 
Ex.  22.  Given  '^(x  —  32)  =  l6  —  ^x,  to  find  the  value  of  ar. 

Ans.  a:=:81. 
Ex.  23.  Given  y(4a;+21)=2Va;-f  1,  to  find  the  value  of 
X.  Ans.  a:=25. 

Ex.24.  Given  ^[l+a;'v/(a;2 4- 12)3=1+3?,  to  find  the  va- 
lue of  a:.  Ans.  a;=2 

36 

Ex.  25.    Given  ■^x+x^{x—9)  =  —r x-,,  to  find  the  va- 

y(a;  — 9) 

lue  of  a?,  Ans.  a? =25. 

Ex.  26.  Given  7  (a-f  a;)=27  [x'^+5ax-^h^),to  find  the  va- 

lue  01  X.  Ans.  x=—- — . 

3a 

Ex.  27.  Given  ^^—r^—-= — +V  ^  ^^  ^^^^j  ^-^^  ^  j^^  of  ^^ 
V«+2        -v/a;+40' 

Ans.  a;=4. 

Ex.  28.  Given    \  f~-=  ^  f~~„,  to  find  the  value  of  x. 
-/ 6x4-2     4/6a;4-6' 

Ans.  x=6. 

Ex.  29.  Given    f"!"^^ /  5^-3    ^^  ^^^  ^^^  ^^j^^ 

/5a?4-3  2       ' 

of  X.  Ans.  a;=5 . 

21 


230  PURE  EQUATIONS. 

Ex.  30.  Given --=zc-{-- ,  to  find  the  value  of  or. 

-/ ax-\-b  c 


Ans..=i.(*+-il)3. 


§  11.   SOLUTION  OF  PURE  EQUATIONS  OF  THE  SECOND,  AND 
OTHER   HIGHER  DEGREES,  BY  EVOLUTION. 

RULE. 

333.  Transpose  the  terms  of  the  equation  in  such  a  man- 
ner, that  the  given  power  of  the  unknown  quantity  may  be 
on  one  side  of  the  equation,  and  the  known  quantities  on  the 
other  ;  then  extract  the  root,  denoted  by  the  exponent  of  the 
power,  on  each  side  of  the  equation,  and  the  value  of  the  un- 
known quantity  will  be  determined.  In  the  same  way  any 
adfected  equation,  having  that  side  which  contains  the  un- 
known quantity,  a  complete  power,  may  be  reduced  to  a  sim- 
ple equation,  from  which  the  value  of  the  unknown  quantity 
will  be  ascertained,  by  the  rules  in  Chap.  III. 

Ex.  1.  Given  a;2— 17=130— 2a;2,  to  find  the  values  of  a:. 

By  transposition,  3ic2=:147  ; 

.-.  by  division,  j:2_49^ 

and  by  evolution,  0:=^  7. 

334.  It  has  been  already  observed,  that  ^y  a  may  be  either 
-h  or  — .  where  n  is  any  whole  number  whatever  ;  and,  con- 
sequently, all  pure  equations  of  the  second  degree  admit  of 
two  solutions.  Thus,  4-7  X  4-7,  and  —  7  x  —7,  are  both 
equal  to  49 ;  and  both,  when  substituted  for  x  in  the  original 
equation,  answer  the  condition  required. 

Ex.  2.  Given  x^-\-ab  =  5x'^^  to  find  the  values  of  x. 

By  transposition,  4x'^—ab  ; 
.-.  2a;=4z'/«^,  and  x=±^^/ab. 
Ex.  3.  Given  x'^  —  6x-{-9=a^,  to  find  the  values  of  a:. 

By  evolution,  ar  — 3 :=:!:«  ;    '.  oc  =  3:^a. 
Ex.  4.  Given  4x'^—4ax-\-d^=x^-\-i2x-\-36,  to  find  the  va- 
lue of  X. 

By  extracting  the  square  root  on  both  sides,  we  have  2x — 
azzix-jrG  ; 

.-.  by  transposition,  x=a-{-6. 
Ex.  5.    Given  ..>+/=13, )  ^^  g^^  ^^^  ^ 

and  x^—p^=5j  y  ^ 

By  addition,  2a;"'^=l 8;  .-.  x—:i^^9=z^3. 
By  subtraction,  2y2_8  ;  ...  y  —  ±J^4  —  ^2. 
Ex.  6.  Given  81x^=256  to  find  the  values  of  x. 


PURE  EQUATIONS.  /         231 

By  extracting  the  square  root,  9a;2  =  -|-,16  ; 
By  extracti:i>g  again,  80?=:^ -y/rt  16  =  ^:4,  or  ±4^^  — 1 ; 

.-.  x=  if,  or  oc=  if-/— 1. 
Ex.  7.  Given  a;^  — 3x*+3a;^  — 1=27,  to  find  the  values  of  a?. 
By  evohition,  a;^  — 1=3  ;  .-.  a;2=4,  and  xzzz-^2. 
Ex.  8.  Given  36x'^=za'^,  to  find  the  values  of  a;. 

Ans.  x=:^:^^a. 
Ex.  9.  Given  a;^=r27,  to  find  the  value  of  a;.  Ans.  x=3. 
Ex.   10.  Given  a;"H-6a;-t-9=25,  to  find  the  values  of  a;. 

Ans.  xz=2,  or  —8. 
Ex.  11.  Given  Sa:^  — 9=21  +  3,  to  find  the  values  of  a:. 

Ans.  a;=i-v/ll. 
Ex.  12.  Given  x^—x'^-\-lx—^Y=a^,  to  find  the  values  of  x. 

Ans.  x=za-i-^. 
Ex.   13.  Given  x^-\-^x-\'^=aH^,  to  find  the  values  of  x. 

Ans.  x=^ah—^. 
Ex.  14.  Given  x^-\'bx-{-jb^=a\  to  find  the  values  of  a:. 

Ans.  a;=  j^a— ^&. 
Ex.  15.  Given  a;*— 2a;2  4-l=9,  to  find  the  values  of  a;. 

Ans.  .rz=_l-2,  or  ±V— 2. 
Ex.   16.  Given  a;*  — 4a;2  4-4=:4,  tofind  the  values  of  a;. 

Ans.  a;=  J^2,  or  4:  v'O. 
Ex.   17.  Given  5a;2—27=3a;2+215,  to  find  the  values  of  a:. 

Ans.  a^=:-|-ll. 
'Ex.  18.  Given  5a;2--l  =244,  to  find  the  values  of  x. 

Ans.  xz=4~7. 
Ex.  19.  Given  9a;24-9  =  3a;24.63,  to  find  the  values  of  x. 

Ans.  a;=  J;3. 
Ex.  20.  Given  2ax'^  +  b—4=cx^—5+d—ax'^,  to  find  the 

values  of  x.  Ans.  a;=  J- .  / 2-. 

V     3a — c 

Ex.  21.  Given  x^-\-y'^^a  and  a;*— y^^^,  to  find  the  values 
of  X  and  y. 

Ans.  a;  =  ±-/(±i-/{2a  +  2^'))  and  y  =  ±y^(±^^(2a  — 
2b)). 

§  III.    EXAMPLES    IN    WHICH    THE    PRECEDING  RULES  ARE  AP- 
PLIED IN  THE  SOLUTION  OF  PURE  EQUATIONS. 

335.  When  the  terms  of  an  equation  involve  powers  of  the 
unknown  quantity  placed  under  radical  signs. 

Let  the  equation  be  cleared  of  radical  signs,  as  in  Sect.  I ; 
then,  the  value  of  the  unknown  quantity  will  be  determined 
by  extracting  the  root,  as  in  Sect.  IL. 

And  by  a  similar  process,  any  equation  containing  the  pow 


232  PURE  EQUATIONS. 

ers  of  a  function  of  the  unknown  quantity,  or  containing  the 
powers  of  two  unknown  quantities,  may  frequently  be  reduced 
to  lower  dimensions.  # 

Ex.  1.  Given  y  x^-^y  («-{-&),  to  find  the  values  of  x. 

Cubing  both  sides,  a;2  =  a-f-5  ; 

.•.a:=iV(a  +  i). 

Ex.  2.  Given^(a;2— 9)=  ■v/(a;-3);  tofindthevaluesofar. 

Here,  the  given  quantity  may  be  exhibited  under  the  form 
JL  '  '  A  X2 

(ir2— 9)4-_(a;_3)'?  ;  then,  by  squaring  both  sides,  (a:^— 9)^ 

=  (a:-3)^^^'  or  (a:2_9)^zra:-3  ; 

by  squaring  again,  x^—^:=zx^ — 6x-t-9; 
.-.by  transposition,  6a;=18  ;  and  ar=:3. 
Ex.  3.  Given  a;^— ^2^9^  and  x—y=L\  ;  to  find  the  values 
of  a;  and  y. 

Dividing  the  corresponding  members  of  the  first  equation 

by  those  of  the  second,  we  have  a?-f-y=9  ; 

adding  this  equation  to  the  second,  2j:=10  ; 

.•.af;=5,  and  y= 9—07 ;  .-.  y=z4. 

Ex.  4.  Given  ^x->r^fy=^,  )  ^  ^^^  ,j^^  ^^j^^^  „f  ^  ^^^ 

and  ya:— vy  =  Ij  S 
Adding  the  two  equations,  2y'a;=:6,  .-.  ■/a::=3, 
and  by  involution,  a:=:9. 
Subtracting  the  two  equations,  2-v/y=4,  and  y'^=2  ; 

.-.by  involution,  y=4. 

Ex.  5.  Given  »^+«y=12.  >  ^  g„j  ,j,,  ^,,„,,  „,  ^  ^^^ 

and  y2H-a:y=:24,  5  ^ 

By  addition,  a;24-2a:y-}-y2— 36  ; 

.*.  extracting  the  square  root,  a?+y=  dt^. 

Now  x^-^xy=.x  .  (a?+3(r)=^6a:  ; 

/. -t-6a:=:  12,  and  a;=:J::2  ; 

Ex.  6.  Given  a;+V(a24.a;2)=:-—— -.,  to  find  the  values 

y(a2 -f-a:2) 

of  X. 

Multiplying  by  -1/(0^4- «^),  we  have  x^/((a?■'\■x'^)-\-a^^\^x'^  — 
2a2; 

by  transposition,  x-y/{c?-\-x'^')^^€?- — a;^, 
and  squaring  both  sides,  c^x'^-\-s^=:a'^—^(P-x^-^x^  ; 

.-.  3a2»2=a*,  and  ac=;i— 7-. 

13 
Ex.  7.  Given  a:2+y2=_- 

^  g  ^  ^  to  find  the  values  of  x  and  y. 
and  a;y= 


PURE  EQUATIONS.  233 

From  the  1st  equation  subtracting  twice  the  2d. 

a;2__2a:y+y2=(a:— y)2=— — ,  .-.  (a?-.-y)3=l, 

and  x—y  =  l  ^  .•.x'^-\-y'^=:l^  ; 
and  2a:y=12  ; 


/.by  addition,  x'^-\-2xy-\-y'^=25, 
.'.  by  extracting  the  square  root,  a:+y=i5  ; 

but  x-%y=l  ; 


Ex.  8.  Given  a,--rv--^io,  f  ,    /•    i  ,u        i  r         j 

to  find  the  values  of  x  and  y 


.'.  by  addition,  2a:=:6,  or— 4  ;  and  a;  =  3,  Or  —2  ; 
by  subtraction,  2y=4,  or— 6  ;  and  y=2,  or  —3. 

and  a;3-j-y^=:  5,  ) 

2         1.  J.      a 
Squaring  the  second  equation,  x^  -{-2x^y^  -\-y^  =i25 

but  j;^  +y^=13 

1  1 

.'.  by  subtraction,  2x^y^=zl2. 
Subtracting  this  from  the  1st  equation, 

x^—2x'^y^-{-y^=l 

.'.  extracting  the  square  root,  x^—y^=z^l 

1        L 
but  a?3-fy3-_5 


i 
.-.  by  addition,  2a!;3  =6,  or  4  ; 
JL 
andx3=3,  or2;    .•.a;r=27,  or  8; 
1. 
.-.by  subtraction,  2y3— 4^  or  6, 

1 
and  y3— 2,  or  3  ;  .'.^=8,  or  27. 

Ex.  9.  Given  x^+x^y^-\-y^=273,  >  to  find  the  values  of  x 

and  x^-j-x"y^-\-y'^—2l,    )  and  y. 
Dividing  the  first   equation  by  the   second,  a;* — x^y^-{-y* 
^13; 

subtracting  this  from  the  second  iequation,  2x'^y^=8  ; 

.'.  x^y'^  =  4: ; 

by  adding  this  equation  to  the  second,  x*-\-2x'^y^-{-y^ 

=25;  .'.x'^+y^=:^5. 

Subtracting  the  equation  x'^y^z=4:,  from  x^^x^y^-\-y^=l3, 

a;*— 2a;2y2+y*=9;  .:x^-'y^=±3, 

21* 


234  .    PURE  EQUATIONS. 

.-.by  addition,  2a;2=-4-8,  and  x—:^2,  or  ±2-/+!  ;  and  by 
subtraction,  2y2— ±2,  and  y  =  ±l,  or  rtV"!- 

Ex.  10.  Given  ^^^±yp:f}-i,  ,o  find  the  value 

of  a?. 

Multiply  the  numerator  and  denominator  by  -v/(^+^)"f"  V 

.'.  y/(a'^—x^)=hx—ai  and  squaring  both  sides,  a^—x^=b^x'^ 
— 2a5a?^a2, 

.-.  b^x^-{-x^=2abx,  and  a?=Y^— — . 

Ex.  11.  Given  (x^—y^)x(x^y)  =  3xy,  K    ^  j  *v        i 
and  (x^--y^)  x(x^-y^)=45xY,  I 
of  a;  and  y. 

Dividing  the  second  equation  by  the  first, 

(oi^-¥y^)  .  (x-{-y)=l5xy  ;   .-.  x^-{-x^y-{-xy^-l-y^—l6xy  ; 

but  from  the  first,     oc^  —  x'^y—xy^-\-y^=   3xy ; 

.-.  by  addition,  2x^+2y^=l8xy,  and  x^-{-y^=9xy. 

But  by  subtraction,  2x'^y-{-2xy'^=l2xy,  and  a;+y  =  6  ; 

.-.  by  cubing,  j^ +30^2^^33,^24.^3  =^216  , 

x^  -{-y^z^dxy  ; 

.-.  by  subtraction,  3x^y-\-3xy^=2l6  —9xy, 
or  3  .{x+y).xy=z3x6  .  xy=2l6-9xy  ;  .-.  27a;y=216,  and 

Now  x^-i-2xy+y^  =  36, 
and  4a:y         =32  ; 

.-.  by  subtraction,  x^—2xy-{-y^=4j 
and  by  extracting  the  square  root,  x—y=^2, 

by  x-{-y=     6, 


.*.  by  addition,  2a;=8,  or  4  ;  and  x=4,  or  2  ; 
and  by  subtraction,  2y  =  4,  or  8  ;  .•.y=2,  or  4. 

Ex.  12.  Given  -+  ^^  ~"'^— ^,  to  find  the  values  of  a?.* 
ac  a?  6 

Ans.  x=:^^(2ab  —  b^). 
1  8 
Ex.    13.    Given  a:2+3a;—7=a:+2H ,  to  find  the  values 

X 

of  X.  Ans.  a; =3,  or  —3. 


V  (a?4-J' 


PURE  EQUATIONS.  235 

to  find  the  values  of  ar.  Ans.  x=— — -rr 

Ex.  15.  Given  x-j-y  :  x  :  :  5  :  3,  and  xi/  =  6,  to  find  the  va- 
lues of  a:  and  y.  Ans.  a;=:i3,  and  y=:jt2. 

Ex.  16.  Given  x—y  :  x  : :  5  :  6,  and  xy^  =  384,  to  find  the 
values  of  x  and  y.  Ans.  a;=24,  and  y=i4. 

Ex.  17.  Given  x-\-i/  :  x  : :  7  :  5,  and  a:y +3/^=126,  to  find 
the  values  of  x  and  y.  Ans.  a;— 4:15,  and  y=±6. 

Ex.  18.  Given  xf+i/=:2l,  and  a'2y*4-y2  — 333^  to  find  the 
values  of  x  and  y.  Ans.  a:=2,  or  jig- ;  and  y  =  3,  or  18. 

Ex.  19.  Given  a;2y+a:y2=  180,  and  a;3+y3= 189,  to  find  the 
values  of  x  and  y.  Ans.  x=5,  or  4  ;  and  y  — 4,  or  5. 

Ex.  20.  Given  x-^-  ^xy-^y=l9,  and  a:2+a:y  +  y2r=  133,  to 
find  the  values  of  x  and  y.       Ans.  a;=9,  or  4  ;  and  y=4,  or  9. 

Ex.  21.  Given  x'^y-{-xy'^=6,  anda;y+*y=12,  to  find  the 
values  of  a;  and  y.  Ans.  a:=:2,  or  1  ;  and  y=:l,  or  2. 

Ex.  22.  Given  {x^-\-y^)  X  (a:-f  y)  =2336,   and  (a:^— y2) 
(a:— y)=576,  to  find  the  values  of  x  and  y. 

Ans.  a;=:l  1,  or  5  ;  and  y=:5,  or  11. 

Ex.  23.  Given  x^-{-y'^z=z(x-\-y)  .  xy,  and  a;2y-|-a?y2=:4a:y,  to 
find  the  values  of  x  and  y.  Ans.  a:=2,  and  y  =  2. 

Ex.  24.   Given  2  .  (a:2-f  y2)  .  (a^+y)  =  15a:y,  and  4  {x'^—y^) 
(a:2-|-y^)  =  75a;2y2j  to  find  the  values  of  x  and  y. 

Ans.  a:=:2,  and  y  =  l 

Ex.  25.  Given  x~y  :  y  :  :  4  :  5,  and  a:2-f-4y2=:181,  tq.find 
the  values  of  x  and  y.  Ans.  x=  ±9,  and  yrr  -[-5. 

Ex.  26.  Given  x'^+y^  :  x^—y^  :  :  17  :  8,  and  a:y2=:45,  to  find 
the  values  of  x  and  y.  Ans.  a:— 5,  and  y  =  3. 

Ex.  27.  Given  ^x— yyz=3,  and  ^  x-\-{^  y=z7 ;  to  find 
the  values  of  a?  and  y.  Ans.  a:  =  625,  and  y=:16. 

Ex.  28.  Given   -y/a^  +  '/y  :  \/x  —  -\/y  :  :  4  :  1,  and  x—y~ 
16,  to  find  the  values  of  x  and  y.  Ans.  a;:=25,  and  y=:9. 

Ex.  29.    Given  a:3+y3  :  x^—y^::  559  :  127,  and  a;2y=294  ; 
to  find  the  values  of  x  and  y.  Ans.  a:=7,  and  y=6. 

Ex.  30.    Given  x^-\-y^=20,  and  x^+y'^=6  ;    to  find  the 
values  of  x  and  y. 

Ans.  a;=-l-8,  or  i  -^8,  and  y=32,  or  1024 
Ex.  31.    Given   a;4+2a:2y2-f-y4  :=  1296— 4a:y(a;2+a:y+y2), 
and  X — y=4  ;  to  find  the  values  of  x  and  y. 

Ans.  5,  or  —1,  and  y=l>  or  —5. 
Ex.  32.  Given    V(^^+1)+V4a;^     ^^  ^^^  ^^^  ^^^^^  ^^  ^^ 
y  (4a;+l)  — V  4aj 

Ans.  x=^f 


236  SOLUTIQN  OF  PROBLEMS 

Ex.  33.  Given  xy  —  d^,  and  a^^+y^— .^2  .  ^o  find  the  values 

of  a;  and  y.  Ans.  x^±\\^J{s'^-\.^a^)J^  y'(^2_2a2)], 

and  y=rl=Mv^(^2_{_2a2)_y(^2_2a2)]. 

Ex.  34,  Given  x^-\-xy  xy^z=:208,  and  y^+y^/  x^y=^l053,  to 
find  the  values  of  a?  and  y.  Ans.  a:=i8,  andy=i27. 

Ex.  35.  Given  x'^+x'^y^-{'y'^=      1009, 

3.    3. 

and  x^  +  x^y^-i-y^  =  582l93, 
to  find  the  values  of  x  and  y. 

Ans.  a;=81,  or  16  ;  and  5/= 16,  or  81. 


CHAPTER  IX. 

ON 

THE    SOLUTION   OF   PROBLEMS, 

PRODUCING  PURE  EaUATIONS. 

336.  In  addition  to  what  has  been  already  said,  with  re- 
spect to  the  translation  of  problems  into  algebraic  equations, 
it  is  very  proper  to  observe,  that,  when  two  quantities  are  re- 
quired which  are  in  the  given  proportion  of  m  to  n,  the  un- 
known quantities  are  represented  by  mx  and  nx ;  then  the 
values  of  a:,  found  from  the  equation  of  the  problem  by  the 
methods  in  the  preceding  chapter,  being  multiplied  by  m  and 
n  respectively,  will  give  the  numbers  required. 

If  three  quantities  are  required,  which  have  given  ratios  to 
one  another,  assume  wia?,  nx,  and  px,  m  to  n  being  the  ratio  of 
the  first  to  the  second,  and  n  to  p  being  that  of  the  second  to 
the  third  ;  then  proceed  as  before. 

Problem  1.  There  are  two  numbers  in  the  proportion  of  4 
to  5,  the  diflference  of  whose  square  is  81.  What  are  those 
numbers  ? 

Let  4a;  and  bx=  the  numbers  ; 
then  (25a:2_i6a;2  =  )  9a:2=81  ;  .*.  x'^=9,  and  a:=  ±3.    Conse- 
quently the  numbers  are  ^^12  and  ±15. 

Prob.  2.  It  is  required  to  divide  18  into  two  such  parts, 
that  the  squares  of  those  parts  may  be  in  the  proportion  of  25 
to  16. 

Let  a;=  the  greater  part ;  then  18— a;  =  the  less  ; 

.-.  a;2  :  (18— a:)2  : :  25  :  16,  and  16x2=25(18— a:)^  ; 


PRODUCING  PURE  EQUATIONS.  237 

.*.  extracting  the  square  root,  4a:  =  5(18— ac),  and 
9a;  =  90  ;  .'.  a;  =^10,  and  the  parts  are  10  and  8. 

Prob.  3.  What  two  numbers  are  those  whose  difference, 
multiplied  by  the  greater,  produces  40,  and  by  the  less  15  ? 
Let  xz=.  the  greater,  and  y=  the  less  ; 
/.  a:2—a:y  =  40,  and  a-y— y^  — 15  ; 
.-.  by  subtraction,  x^ — 2a:y4-y2— 25, 

and  x—yz^^b. 
.'.  from  the  first  equation,  x[x — y)  =  J::5a;=40, 

and  x=.-^S. 
From  the  2d,  y(a:— y)=r±5y=±15  ;  .-.  y  =  ±3.    * 

Prob.  4.    What  two  numbers  are  those  whose  difference, 
multiplied  by  the  less,  produces  42,  and  by  their  sum  133  ? 
Let  x=:  the  greater,  and  y=  the  less  ; 

/.  {x—y)  .  y=:42,  and  {x^y)  .  (a:4-y)  =  133  ; 
/.iy  subtracting  twice  the  first  from  the  second, 
a;2 — 2j:y+y2=49  ;  .-.  a:— y=-l-7  ; 
whence  -l-7y  =  42,  and  y  =  ±6  ; 
buta;=yi7;  .-.  a;=i6±7  =  d=13. 

Prob.  5.  What  two  numbers  are  those,  which  being  both 
multiplied  by  27,  the  first  product  is  a  square,  and  the  second 
the  root  of  that  square  ;  but  being  both  multiplied  by  3,  the 
first  product  is  a  cube,  and  the  second  the  root  of  the  cube  ? 
Let  X  and  y  be  the  numbers  ; 

then  -^'llx  —  Tly,  and  .*.  x=21y'^ , 
also  -y  3a;  =  3y  ;  and  .-.  x—9y^  ; 
whence  9y3=:27y2,  and  y=3  ;  .-.  a;=9  X  27=243  ; 
.'.  the  numbers  are  243,  and  3. 

Prob.  6.  Two  travellers,  A  and  B,  set  out  to  meet  each 
other  ;  A  leaving  the  town  C  at  the  same  time  that  B  left  D. 
'rfiey  travelled  the  direct  road,  CD  :  andy  on  meeting,  it  ap- 
peared that  A  had  travelled  18  miles  more  than  B  :  and  that 
A  could  have  gone  B's  journey  in  15|^  days,  but  B  would  have 
been  28  days  in  performing  A's  journey.  What  was  the  dis- 
tance between  C  and  D  1 

Let  a;=  the  number  of  miles  A  has  travelled  ; 

.'.ar— 18=  the  number  B  has  travelled  ; 

and  a;  — 18  :  a;  :  :  15J  :  the  number  of  days  A  travelled,  = 

63a; 
— — - ;  also  a; :  a;— 18  : :  28  :  to  the  number  of  days  B  tra- 

18)2=9a;2;  .-.4.(3;— 18)  =  i3a?,  and  a;=72,  or  lOf  ;  whence 


238  SOLUTION  OF  PROBLEMS 

A  travelled  72,  and  B  54  miles  ;  and,  the  whole  distance,  CD 
126  miles. 

Prob.  7.  Two  partners,  A  and  B,  dividing  their  gain  (60/), 

B  took  20/.     A's  money  continued  in  trade  4  months ;  and  if 

the  number  50  be  divided  by  A's  money,  the  quotient  will  give 

the  number  of  months  that  B's  money,  which  was  100/.,  con- 

^  tinned  in  trade.     What  was  A's  money,  and  how  long  did  B's 

money  continue  in  trade  ? 

50 
Suppose  A's  money  was  x  pounds  ;  .*.  — =  the  number  of 

months  B's  money  was  in  trade  ;  and  since  B  gained  20/.,  A 

gained  40/. 

,       50X100       ^     ,         ,  ,        10000 
/.  4:X  : :  :  2  :  1,  and  4x=: ; 

X  X 

.:  4a;2=:10000,  and  a:2=2500  ;  .-.  a;=±50. 
.*.  A's  money  was  50/.,  and  B's  money  was  one  month  in 
trade. 

Prob.  8.  A  detachment  from  an  army  was  marching  in  re- 
gular column,  with  5  men  more  in  depth  than  in  front ;  but 
upon  the  enemy  coming  in  sight,  the  front  was  increased  by 
845  men  ;  and  by  this  movement  the  detachment  was  drawn 
up  in  five  lines.     Required  the  number  of  men. 

Let  x=  the  number  in  front ; 
.••  a:+5=  the  number  in  depth, 
and  x{x-^5)=z  the  whole  number  of  men  ; 
also,  (a:-f-845)x5=:  the  whole  number  of  men  ; 
.-.  a:2-f  5a;=5a:  +  4225,  and  a;2  =  4225  ;  .-.  a;=±65. 

And,  consequently,  5a:4-4225  —  325+4225  =  4550,  the 
number  of  men.  Here,  although  the  negative  value  of  x  will 
not  answer  the  conditions  of  the  problem,  yet  it  will  satisfy 
the  above  equation  ;  for,  if  we  substitute  — 65  for  x,  we  shall 
have  (  — 65)2  +  5(  — 65)=5(  — 65)  +  4225  ;  that  is,  or  4225  — 
325  =  — 325  +  4225  ;  .-.  4225=4225,  or  4225—4225=0,  that 
is,  0=0. 

Prob.  9.  It  is  required  to  divide  the  number  a  into  two  such 
parts,  that  the  squares  of  those  parts  may  be  in  the  proportion 
of  m  to  n. 

Let  x=  one  of  these  parts  ;  then  a — x=z  the  other  ;  and  ac- 
cording to  the  enunciation  of  the  problem,  we  shall  have  the 
equation, 

x^         m  X  ,     /^  /     ,r       ^         f\  t. 

— =— :  .-. =  +^/ — ,  or  (puttmg — =m),x  =  ± 

(a-x)2      n'       a-x     "^^jn'        ^^        ^n  " 

{a—x)^/mf. 


PROTDUCING  pure  equations.  239 

By  resolving  separately  the  two  equations  of  the  first  de- 
gree comprised  in  the  above  formula,  namely, 

x=-j-{a—x)^m',  and  x=—-(a~-x)-\/m'j 
we  shall  have,  from  the  first, 

—  and  from  the  second  x: 


1  +  Vm"  """  """' """  """"""  —  1-  ^m'  • 
By  the  first  solution,  the  second  part  of  the  proposed  num- 

ber  IS  a-  -^— —--———  ;  and  the  two  parts,    ^ 

1+ym       \-\-y'm  1  +  ym' 

and  -— — —  -,  are,  as  was  required  in  the  enunciation  of  the 

1+  yjmf 

question,  both  less  than  the  number  proposed- 
By  the  second  solution,  we  have 

(~a\/m'\          ,     a'J mf               x  ,    , 

; --—r\=a-\-^ —     ,    ■  =  - ;— 7 ;  and  the  two  parts 
l  —  ^m/            l—y/m      X—^nv 

aWm  -         ^ 

are  —  -— ^-— — -,  and 


1  —  ^m^  1  — -y/m'* 

Their  signs  being  contrary,  the  number  a  is  not,  properly 
speaking,  their  sum,  but  their  difference. 

Now,  if  a=18,  m=25,  and  7i=l6  ;  then  substituting  these 

values  in  the  formula  — - — ; — -,  and  -—; — ; — -.,  we  shall  find  10 

l-{-y/m^         l+V^ 
and  8  equal  to  the  two  parts  required,  the  same  as  in  Ex.  2., 
which  is  a  particular  case  of  this  general  problem. 

•  Prob.  10.  What  two  numbers  are  those,  whose  sum  is  to 
the  greater  as  10  to  7  ;  and  whose  sum,  multiplied  by  the 
less,  produces  270?  Ans.  ±21  and  ^9. 

Prob.  11.  What  two  numbers  are  those,  whose  difference 
is  to  the  greater  as  2  to  9,  and  the  difference  of  whose  squares 
is  128?  Ans.  ^18  and  ±14. 

Prob.  12.  A  mercer  bought  a  piece  of  silk  for  16/.  4^. ; 
and  the  number  of  shillings  which  he  paid  for  a  yard  was  to 
the  number  of  yards  as  4  :  9.  How  many  yards  did  he  buy, 
and  what  was  the  price  of  a  yard  ? 

Ans.  27  yards,  at  126-.  per  yard. 
Prob.  13.  Find  three   numbers   in  the  proportion  of  ^,  |-, 
and  J  :  the  sum  of  whose  squares  is  724. 

Ans.  ±12,  i  16,  and  ±18. 
Prob.  14.  It  is  required  to  divide  the  number  14  into  two 


240  SOLUTION  OF  PROBLEMS. 

such  parts,  that  the  quotient  of  the  greater  part,  divided  by 
the  less,  may  be  to  the  quotient  of  the  less  divided  by  the 
greater  as  16  :  9.  Ans.  The  parts  are  8  and  6. 

Prob.  15.  What  two  numbers,  are  those  whose  difference 
is  to  the  less,  as  4  to  3  ;  and  their  product,  multiplied  by  the 
less,  is  equal  to  504  ?  Ans.  14  and  6. 

Prob.  16.  Find  two  numbers,  which  are  in  the  proportion 
of  8  to  5,  and  whose  product  is  equal  to  360. 

Ans.  ±24,  and  ±15. 

Prob.  17.  A  person  bought  two  pieces  of  linen,  which, 
together^  measured  36  yards.  Each  of  them  cost  as  many 
shillings  per  yard,  as  there  were  yards  in  the  piece  ;  and  their 
whole  prices  were  in  the  proportion  of  4  to  1.  What  were 
the  lengths  of  the  pieces  ?  Ans.  24  and  12  yards. 

Prob.  18.  There  is  anumber  consisting  of  two  digits,  which 
being  multiplied  by  the  digit  on  the  left  hand,  the  product  is 
46  ;  but  if  the  sum  of  the  digits  be  multiplied  by  the  same 
digit,  the  product  is  only  1 0.     Required  the  number. 

Ans.  23. 

Prob.  19.  From  two  towns,  C  and  D,  which  were  at  the 
distance  of  396  miles,  two  persons,  A  and  B,  set  out  at  the 
same  time,  and  met  each  other,  after  travelling  as  many  days 
as  are  equal  to  the  difference  of  the  number  of  miles  they  tra- 
velled ^er  day;  when  it  appears  that  A  has  travelled  216 
miles.     How  many  miles  did  each  travel  ■per  day  ? 

Ans.  A  went  36,  and  B  30. 

Prob.  20.  There  are  two  numbers,  whose  sum  is  to  the 
greater  as  40  is  to  the  less,  and  whose  sum  is  to  the  less  as  90 
is  to  the  greater.     What  are  the  numbers  ? 

Ans.  36,  and  24. 

Prob.  21.  There  are  two  numbers,  whose  sum  is  to  the 
less  as  5  to  2  ;  and  whose  difference,  multiplied  by  the  dif- 
ference of  their  squares,  is  135.     Required  the  numbers. 

Ans.  9,  and  6. 

Prob.  22,  There  are  two  numbers,  which  are  in  the  pro- 
portion of  3  to  2  ;  the  difference  of  whose  fourth  powers  is  to 
the  sum  of  their  cubes  as  26  to  7.     Required  the  numbers. 

Ans.  6,  and  4. 

Prob.  23.  A  number  of  boys  set  out  to  rob  an  orchard, 
each  carrying  as  many  bags  as  there  were  boys  in  all,  and 
each  bag  capable  of  containing  4  times  as  many  apples  as 


PRODUCING  PURE  EQUATIONS.  241 

• 
there  were  boys.     They  filled  their  bags,  and  found  the  num- 
ber of  apples  was  2916.     How  many  boys  were  there  ? 

Ans.  9  boys. 

Prob.  24.  It  is  required  to  find  two  numbers,  such  that  the 
product  of  the  greater,  and  square  of  the  less,  may  be  equal 
to  36  ;  and  the  product  of  the  less,  and  square  of  the  greater, 
may  be  48.  Ans.  4,  and  3. 

Prob.  25.  There  are  two  numbers,  which  are  in  the  pro- 
portion of  3  to  2  ;  the  difference  of  whose  fourth  powers  is  to 
the  difference  of  their  squares  as  52  to  1.  Required' the  num- 
bers. Ans.  6,  and  4. 

Prob.  26.  Some  gentlemen  made  an  excursion,  and  every 
one  took  the  same  sum.  Each  gentleman  had  as  many  ser- 
vants attending  him  as  there  were  gentlemen  ;  and  the  num- 
ber of  dollars  which  each  had  was  double  the  number  of  all 
the  servants  ;  and  the  whole  sum  of  money  taken  out  was 
$3456.     How  many  gentlemen  were  there  ?  Ans.  12. 

Prob.  27.  A  detachment  of  soldiers  from  a  regiment,  be- 
ing ordered  to  march  on  a  particular  service,  each  company 
furnished  four  times  as  many  men  as  there  were  companies 
in  the  regiment ;  but  those  becoming  insufficient,  each  com- 
pany furnished  3  more  men  ;  when  their  number  was  found 
to  be  increased  in  the  ratio  of  17  to  16.  How  many  compa- 
nies were  there  in  the  regiment  ?  Ans.  12. 

Prob.  28.  A  charitable  person  distributed  a  certain  sum 
among  some  poor  men  and  women,  the  numbers  of  whom 
were  in  the  proportion  of  4  to  5.  Each  man  received  one- 
third  of  as  many  shillings  as  there  were  persons  relieved  ; 
and  each  woman  received  twice  as  many  shillings  as  there 
were  women  more  than  men.  Now  the  men  received  all  to- 
gether 185.  more  than  the  women.  How  many  were  there 
of  each  ?  Ans.  12  men,  and  15  women. 

Prob.  29.  Bought  two  square  carpets  for  62/.  1^. ;  for  each 
of  which  I  paid  as  many  shillings  per  yard  as  there  were  yards 
in  its  side.  Now  had  each  of  them  cost  as  many  shillings 
per  yard  as  there  were  yards  in  a  side  of  the  other,  I  should 
have  paid  lis.  less.     What  was  the  size  of  each  ? 

Ans.  One  contained  81,  and  the  other  64  square  yards. 

Prob.  30.  A  and  B  carried  100  eggs  between  them  to  mar- 
ket, and  each  received  the  same  sum.  If  A  had  carried  as 
many  as  B,  he  would  have  received  18  pence  for  them  ;  and 
if  B  had  only  taken  as  many  as  A,  he  would  have  received  8 
pence.     How  many  had  each  1  Ans.  A  40,  and  B  60 

22 


242  QUADRATIC    EQUATIONS. 

Prob.  31.  The  sum  of  two  numbers  is  5  (s),  and  their  pro- 
duct 6  (p) :  What  is  the  sum  of  their  5th  powers  ? 

Ans.  275  =  {s^  —  5ps^-{-5p^s), 


CHAPTER  X, 


ON 


QUADRATIC    EQUATIONS. 

337.  Quadratic  equations,  as  has  been  already  observed, 
are  divided  into  pure  and  adfected.  All  pure  equations  of 
the  second  degree  are  comprehended  in  the  formula  x^=ny 
where  n  may  be  any  number  whatever,  positive  or  negative^ 
integral  or  fractional.  And  the  value  of  x  is  obtained  by  ex- 
tracting the  square  root  of  the  number  n  ;  this  value  is  dou- 
ble, for  we  have,  x=z^-y/n,  and  in  fact,  {^-y/nYz^n.  This 
may  be  otherwise  explained,  by  observing,  (Art.  106),  that 
x'^  —  n=z(x-\-^/n).{x  —  ^/n):=^o,  and  that  any  product  consist- 
ing of  two  factors  becomes  nought,  when  there  is  no  restric- 
tion in  the  equality  to  zero  of  that  product,  by  making  each 
of  its  factors'  equal  to  zero. 

We  have,  therefore,  x=z  —  ^/ny  x:=:z-\--y/n,  or  x=i^-\/n. 

338.  Now,  since  the  square  root  is  taken  on  both  sides  of 
the  equation,  x'^=-n,  in  order  to  arrive  at  x=^  -/n  ;  it  is  very 
natural  to  suppose  tkat,  x  being  the  square  root  of  a;^,  we 
should  also  affect  x  with  the  double  sign  ^  ;  and,  therefore,  in 
resolving  the  equation  x^=n,  we  should  write  iaj^iy'n  ; 
but  by  arranging  these  signs  in  every  possible  manner,  namely; 

-{-x=-\-  -^n,   -]-x=z  —  -y/n, 

— x=  —  -\/7i,    — X— -f--v/n, 
we  would  still  have  no  more  than  the  two  first  equations,  that 
is,  4-^=^1/^  i  for  if  we  change  the  signs  of  the  equations 
■^x=-~'\/n  and  — a:=4-  y/n,  they  become  +a;=  +  ^»  and 
-{■xc=z  —  '}/n,  or  x=-^  y/71. 

339.  If,  in  the  formula  x'^=n,  n  be  negative,  or,  which  is  the 
same  thing,  if  we  have  x^=—n,  where  n  is  positive  ;  then, 
«=±V— n=±VwX  V  — 1»  and  in  fact(±  V'»)^X('/— 1)* 


QUADRATIC  EQUATIONS.  243 

=nX  — 1  =  — n  ;  therefore,  the  two  roots  of  a  pure  equation 
are  either  both  real  or  both  imaginary. 

340.  All  adfected  quadratic  equations,  after  being  properly 
reduced  according'to  the  rules  pointed  out  in  the  reduction  of 
simple  equations,  may  be  exhibited  under  the  following  general 
forms;  namely,  a;2-j-na:=:o,  and  a;2-}-na:=:n' ;  where  n  and  w' 
may  be  any  numbers  whatever,  positive,  or  negative,  integral 
ox  fractional. 

341.  The  solution  of  adfected  quadratic  equations  of  the 
ioxmx^+nx^o,  is  attended  with  no  difficulty  ;  for  the  equation 
a;2-|-na;=o,  being  divided  by  x,  becomes  x-\-n=^o,  from  which 
we  lind  a;= — n,  though  we  find  only  one  value  of  x,  according 
to  this  mode  of  solution,  still  there  may  be  two  values  of  x, 
which  will  satisfy  the  proposed  equation. 

In  the  equation,  a;2=:3a?,  for  example,  in  which  it  is  required 
to  assign  such  a  value  of  x,  that  x"^  may  become  equal  to  3a?, 
this  is  done  by  supposing  a:==3,  a  value  which  is  found  by  di- 
viding the  equation  by  a? ;  but  besides  this  value^  there  is  also 
%nother  which  is  equally  satisfactory  ;  namely,  x=.o  ;  for  then 
ti^-=.o,  and  3a:  =  0. 

342.  An  adfected  quadratic  equation  is  said  to  be  complete, 
when  it  is  of  the  form  x'^-\-nx-=zn' ;  that  is,  when  three  terms 
are  found  in  it ;  namely,  that  which  contains  the  square  ol 
the  unknown  quantity,  as  x  ;  that  in  which  the  unknown  quan- 
tity is  found  only  in  the  first  power,  as  nx  ;  and  lastly,  the  term 
which  is  composed  only  of  known  quantities  ;  and,  as  there 
is  no  difficulty  attending  the  reduction  of  adfected  quadratic 
equations  to  the  above  form  by  the  known  rules  :  the  whole 
is  at  present  reduced  to  determining  the  true  value  of  x  from 
the  equation  x^-\-nxz=^n' . 

We  shall  begin  with  remarking,  that  if  a^^+na:  were  a  real 
square,  the  resolution  would  be  attended  with  no  difficulty,  be- 
cause that  it  would  be  only  required  to  extract  the  square  root 
on  both  sides,  in  order  to  reduce  it  to  a  simple  equation. 

343.  But  it  is  evident  that  x^-^-nx  cannot  be  a  square  ;  since 
we  have  already  seen,  that  if  a  root  consists  of  two  terms,  for 
example,  x-\-a,  its  square  always  contains  three  terms,  namely, 
twice  the  product  of  the  two  parts,  besides  the  square  of  each 
part,  that  is  to  say,  the  square  of  x-{-a  \%^x^-\-1ax-\-a^. 

344.  Now,  we  have  already  on  one  side  x^-{-nx  ;  we  may, 
therefore  consider  x^  as  the  square  of  the  first  part  of  the  root, 
and  in  this  case  nx  must  represent  twice  the  product  of  x,  the 
first  part  of  the  root,  by  the  second,  part :  consequently,  this 
second  part  must  be  \n,  and  in  fact  th*e  square  oix\i\n  is  found 
to  be  x^-\-nx-\-\n^. 


244  QUADRATIC  EQUATIONS. 

345,  Now  x'^-{-nx-\-^n'^,  being  a  real  square,  which  has  for 
•its  root  x+ Jti,  if  we  resume  our  equation  x'^-j-nx=n',  we  have 
only  to  add  ^ri^,  to  both  sides,  which  gives  us  x^-\-?ix-\-^n'^  =  n 
4-Jw2,  the  first  side  being  actually  a  square,  and  the  other 
containing  only  known  quantities.  If,  therefore,  we  take  the 
square  root  of  both  sides,  we  find  x-\-^n=  ^ {^ri^-^n)  ;  and 
as  every  square  root  may  be  taken  either  affirmatively  or  ne- 
gatively, we  -shall  have  for  x  two  values  expressed  thus  ; 

x=-^in4:v^  (in^  +  n^y 

346.  This  formula  contains  the  rule  by  which  all  quadratic 
equations  may  be  resolved,  and  it  will  be  proper,  as  Euler 
justly  observes,  to  commit  it  to  memory,  that  it  may  not  be 
necessary  to  repeat,  every  time,  the  whole  operation  which  we 
have  gone  through.  We  may  always  arrange  the  equation  in 
such  a  manner,  that  the  pure  square  x^  may  be  found  on  one 
side,  and  the  above  equation  have  the  form  x^=L—jix-\-n'y 
where  we  see  immediately  that  x=:z — Jn±y/  {^n^-^-n'). 

347  The  general  rule,  therefore,  which  may  be  deduced 
from  that,  in  order  to  resolve  x'^=—nx-\-n\  is  founded  on  thin 
consideration.  That  the  unknown  x  is  equal  to  half  the  co- 
efficient or  multiplier  of  x  on  the  ^ther  side  of  the  equation, 
plus  or  minus  the  root  of  the  square  of  this  number,  and  the 
known  quantity,  which  forms  the  third  term  of  the  equation. 

Thus,  if  we  had  the  equation  x^=6x-{-7,  we  should  imme- 
diately say,  that  a?=:3±y'(9-|-7)  =  3±4  ;  whence  we  have 
these  two  values  of  a; ;  namely,  x-—7,  and  x=  —  l. 

348.  The  method  of  resolving  adfected  quadratic  equations 
will  be  still  better  understood  by  the  four  following  forms  ;  in 
which  n  and  n'  may  be  a^ny positive  numbers  whatever,  integral 
or  fractional. 

I.  In  the  case  x^-\-nx=n',  where  x=:—^n-\-  ■^{^n'^-\-7i'),or 
— J'l — V{i^^~^^')^  ^^®  ^^^^  value  of  x  must  be  positive,  be- 
cause '\/{\n'^-{-n')  is  >-v/j«^,  or  its  equal  Jn  ;  and  its  second 
value  will  evidently  be  negative,  because  each  of  the  terms  of 
which  it  is  composed  is  negative. 

II.  In  the  case  x'^  —  nx=:n\  from  which  we  find  x=^n-{-^ 
(Jn^-j-n'),  or  -Jn— -/(i^^  +  ^O'  ^^®  first  value  of  a:,  is  manifest- 
ly positive,  being  the  sum  of  two  positive  terms ;  and  the  se- 
cond value  will  be  Jiegative,  because  •\/(i'*^"i~'* )  ^^  ^  V(i'*^)» 
or  its  equal  ^n. 

III.  In  the  case  x'^—nx=:  —n\  we  have  x  —  \n-\-  -/(in^— n'), 
or  Jn  — -v/(Jn2— n^) ;  both  the  values  of  x  will  be  positive, 
when  1^2  is  >n' ;  for  its  first  value  is  then  evidently  positive, 
being  composed  of  two  positive  terms  ;  and  its  second  value 
will  also  be  positive,  because -/(i^^—^O  is  less  than/  (J'*")* 


QUADRA.TIC  EQUATIONS.  245 

or  its  equal  Jn.  But  if  J-n^,  in  this  case,  be  less  than  n%  both 
the  values  of  x  will  be  imaginary ;  because  the  quantity,  \n^ 
—  n'  under  the  radical  sign,  is  then  negative  ;  and  consequently 
'^(\ji^  —  n')'wi\\  be  imaginary,  or  of  no  assignable  value. 

IV.  Also,  in  the  fourth  case,  x^-^nx^i — n\  where  x  =. — 
\n-{-  ^J[\n'^—n'),  or  — J«—  VCi^^""^')'  ^^^  ^^^  values  of  x 
will  be  both  negative,  or  both  imaginary,  according  as  ^n^  is 
greater  or  less  than  n'. 

349.  Hence  wt#may  conclude,  from  the  constant  occur- 
rence of  the  double  sign  before  the  radical  part  of  the  preced- 
ing expressions,  that  every  quadratic  equation  must  have  two 
roots ;  which  are  both  real,  or  both  imaginary  ;  and  though 
the  latter  of  these  cannot  be  considered  as  real  quantities,  but 
merely  as  pure  algebraic  symbols,  of  no  determinate  value,_ 
yet  when  they  are  submitted  to  the  operations  indicated  by  the 
equation,  the  two  members  of  that  equation  will  be  always 
identical,  or  which  is  the  same,  it  shall  be  always  reduced  to 
the  form  0  —  0 

350.  It  may  here  also  be  further  observed,  that,  in  some 
equations  involving,  radical  quantities  of  the  form  -y/i^ax-^-b) 
both  values  of  a;,  found  by  the  ordinary  process,  nvill  not  an- 
swer the  proposed  equation,  except  that  we  take  the  radical 
quantity  with  the  double  sign  -j^.  Let,  for  example,  the  values 
of  X  be  found  in  the  equation  x^\-^/{6x-{-\0)zzzS. 

Here,  by  transposition,  y/ (bx-\-lO)^=zS—x , 
therefore  by  squaring,  5a;+10=:64  —  IQx-^-x"^, 
ox  x^—2\x= — 54;  and  •-.  a:=18,  or  3. 
Now,  since  these  two  values  of  x  are  found  from  the  reso- 
lution of  the  equation  a;^— 2]a;=  — 54  ;  it  necessarily  follows, 
that  each  of  them,  when  substituted  for  x,  must  satisfy  that 
equation ;  which  may  be  verified  thus  ;  in  the  first  place,  by 
substituting  18  for  x,  in  the  equation  x'^—2\x=z — 54,  we  have 
(18)2  —  21  Xl8ri:— 54,  or  324— 378=— 54  ;   that  is,  —54  = 
—54,  or  0  =  0. 

Again,  substituting,  3  for  x,  we  have  (3)2— 21  X3  =  — 54, 
or  9-63  =  — 54;   — 54= -54,  or  54— 54=0  ;  .-.0  =  0. 

351.  And  as  the  equation  ot^- 21a;=— 54,  may  be  deduced 
from  the  equation  +-/ (5a;-|-10)  =  8— a?,  or  — /(5a:+10)  = 
8— a? ;  it  is  evident  that  the  radical  quantity-/  (5a:-(-10),  must 
be  taken,  with  the  double  -sign  ^^ ,  in  the  primitive  equation, 
in  order  that  it  would  be  satisfied  by  the  values,  18  and  3,  oi 
X,  above  found  ;  that  is,  18  answers  to  the  sign  — ,  and  3  to 
the  sign  +  •  But  if  one  of  these  signs  be  excluded  by  the 
nature  of  the  question  ;  then  only  one  of  the  values  will  sa- 

22* 


246  QUADRATIC  EQUATIONS. 

tisfy  the  original  equation  ;  for  instance,  if  in  the  equation  x 
4--/ (5a; +10)  =  8,  the  sign  —  be  excluded  from  the  radical 
quantity,  then  the  square  root  of  5a;-f  10  must  be  considered 
as  a  positive  quantity  ;  and  because  it  is  equal  to  8  —  x;  the 
value  of  oc,  since  both  are  positive,  which  will  answer  the  pro- 
posed equation,  must  be  less  than  8  ;  therefore,  3  is  the  value 
of  X,  whioh  will  satisfy  the  equation  x-\-y^  (5a:+10)z=8,  which 
can  be  readily  verified  thus  ;  substituting  3  for  a:,  we  have  3  + 
V(15-|-10)=:8,  or  3  +  5=18.  And  for  a  stnilar  reason,  18  is 
the  value  of  x,  which  will  answer  the  equation  a;— y'^5a;+10) 
=8;  for  18- -v/(90  +  10)  =  18  — 10=:8;  /.  8  =  8,  or  0=0. 

352.  It  is  proper  to  take  notice  here  of  the  following  me- 
thod of  resolving  quadratic  equations,  the  principle  of  which 
is  given  in  the  Bija  Ga?iita,  before  mentioned  :  thus,  if  a 
quadratic  equation  be  of  the  form  4a'^x^4^4abx=J^4ac,  it  is 
evident  that,  by  adding  b^  to  both  sides,  the  left-hand  member 
will  be  a  complete  square,  since  it  is  the  square  of  2«a:  +  Z>  ; 
and,  therefore,  by  extracting  the  square  root  of  both  sides,  there 
will  arise  a  simple  equation,  from  which  the  values  of  x  may 
be  determined. 

353.  Now,  any  quadratic  equation  of  the  form  ax^dcbx=: 
+  c,  (to  which  every  quadratic  may  be  reduced  by  the  known 
rules),  by  multiplying  both  sides  by  4a,  will  become  4rtV"^i 
4a6a;=+4ac.  From  which  we  infer,  that  if  each  side  of  the 
equation  be  muUipUed  by  four  times  the  coejfficient  of  x^^  and  to 
each  side  there  be  added  the  square  of  the  coefjicient  of  x,  the  quan- 
tity on  the  left-hand  side  of  the  resulting  eqtiation  will  always  be 
a  complete  square  ;  from  vihich,  by  extracting  the  square  root,  the 
values  of  X  will  be  determined.  If  the  coefficient  a=l,  then 
both  sides  of  the  equation  is  multiplied  by  4,  and  the  square  of  the 
coefficient  of  x  is  added,  as  before. 

§  I.    SOLUTION   OF  ADFECTED  QUADRATIC   EQUATIONS,  INVOLV- 
ING ONLY  ONE  UNKNOWN  QUANTITY. 

354.  Rule  I.  Let  the  terms  be  arranged  on  one  side  of 
the  equation,  according  to  the  dimensions  of  the  unknown 
quantity,  beginning  with  the  highest ;  and  the  known  quanti- 
ties be  transposed  to  the  other  ;  then,  if  the  square  of  the  un- 
known quantity  has  any  coefficient,  either  positive  or  negative, 
let  all  the  terms  be  divided  by  this  coefficient.  If  the  square 
of  half  the  coefficient  of  the  second  term  be  now  added  to  both 
sides  of  the  equation,  that  side  which  involves  the  unknown 
quantity  will  become  a  complete  square ;  and  extracting  the 


QUADRATIC  EQUATIONS.  247 

square  root  on  both  sides  of  the  equation,  a  simple  equation 
will  be  obtained,  from  which  the  values  of  the  unknown  quan- 
tity may  be  determined. 

355.  Rule  IL  The  terms  of  the  equation  being  arranged 
as  above,  let  each  side  be  multiplied  by  four  times  the  coeffi- 
cient of  x^,  and  to  each  side  add  the  square  of  the  coefficient 
of  X ;  then  the  left-haHd  member,  being  a  complete  power,  ex- 
tract the  square  root  on  each  side  of  the  equation,  and  there 
arises  a  simple  equation,  from  which  the  values  of  x  may  be 
determined. 

356.  It  may  be  observed,  that  all  equations  may  be  solved 
as  quadratics,  by  completing  the  square,  in  which  there  are 
two  terms  involving  the  unknown  quantity,  or  any  function  of 
it,  and  the   exponent   of  one   is   double   that   of  the   other- 

Thus,  x^-\-px^=q,  x^''—paf=q,  x^-\-x*  =  a,  a^x'^-\-ax=b,  x 

-\-ax"^=ih,  /jV*"— J9x2''=<f,  {x^^-px-^-qY-^^x^-^'-px  ^  q)^r, 
a:.(a;4  aa:)2  4-&a:.(a!;2-f-aa:)=c?,areof  the  same  form  as  quadratics, 
and  the  values  of  the  unknown  quantity  may  be  determined 
in  the  same  manner. 

357.  Many  equations  also,  iri  which  more  than  one  unknown 
quantity  are  involved,  may,  in  a  similar  manner,  be  reduced  to 
lower  dimensions  by  completing  the  square,  as  x^y'^-\-pxy:=q, 
(a;3-f-y3)24-jo.(a;^+y^)=r.  Instances  of  this  kind  will  occur 
in  the  next  section. 

358.  And  many  adfected  equations  of  the  third,  and  other 
higher  degrees,  may  be  exhibited  under  the  form  of  a  quad- 
ratic, from  which,  by  completing  the  square,  the  value  of  the 
unknown  quantity  will  be  determined.  The  biquadratic  equa- 
tion x^—Sax^-^Sa'^x^-\-Z2a^x  =  d,  for  instance,  may  be  reduced 
to  the  form  (x^ — Aaxf — Sa^x"^  —  4ax)  —  d.  Thus  the  two  first 
terms  {x^ — 4ax)  of  the  square  root  of  the  left-hand  member 
being  found  according  to  the  rule  (Art.  299),  and  the  remain- 
der —  8a^x'^-\-32a^x,  being  evidently  equal  to  — 8a^{x^ — 4ax) ; 
therefore  x*  —  8ax^-\-8a'^x'^-\-32a^x={x'^ — Aax)"^ — 8a^{x^ — 4ax) 
=  d.  Hence  it  follows,  that  if  the  remainder,  after  having 
found  the  first  two  terms  of  the  square  root  (Art.  238),  can  be 
resolved  into  two  factors,  so  that  the  factor  containing  the  un- 
known quantity,  shall  be  equal  to  the  terms  of  the  root  thus 
found ;  the  proposed  biquadratic  may  always  be  reduced  to  a 
quadratic  form. 

359.  In  a  similar  manner,  the  cubic  equation  x^-{-2ax^-\~ 
5a^x-\~4a^=o,  may  be  reduced  to  the  form  (x^+aa:)^  x4a2 
{x^-\-ax)=o  ;  thus,  multiplying  every  term  of  the  proposed 


248  QUADRATIC  EQUATIONS. 

equation  by  x,  it  becomes  x^ -{-^ax^ -\-ba'^x^ -\-Aa^x=zO,  which 
can  be  reduced  to  the  above  form,  as  in  the  preceding  article. 
There  are  a  variety  of  other  artifices  for  reducing  equations 
to  lower  dimensions,  which  will  be  illustrated  in  the  following 
examples. 

Ex.   1.  Given  x'^-\~Sx  —  20,  to  find  the  values  of  x. 
Completing  the  square,  a;2-f8a;4- 16  —  36  ; 

and  extracting  the  root,  x-\-A,=z^Q  ; 
Whence,  by  transposition,  a; =2,  or  — 10. 
Ex.  2.  Given  a:^— 8a; +5  =  14,  to  find  the  values  of  ». 

By  transposition,  a;^— 8a;=9  ; 
and  completing  the  square,  a;2  —  8a?4- 16=25  ; 
.-.extracting  the  root,  a;— 4=1^5, 
and  a;=9,  or  —1. 

Ex.  3.   Given  "'^^j'f'Z^l-(x-2Y,  to  find   the  values 
a:— -Y/(a;2— 9)      ^ 

of  X. 

Multiplying  the  numerator  and  the  denominator  of  the  fraction 

hy  x-]-^(x^—9y- ^ '-^:=(a:-2)2  ;  .-. ^-^ '-, 

=  i(^— 2) :  Taking  the  positive  sign,  a;-|--\/(a;^— 9)  =  3a;  — 6. 
or  V'(a;2— 9)  =  2a;— 6  ;  .-.  a;^  — 9  =  4a:2— 24a;+36  ;  by  transpo- 
sition and  division,  a:^  — 8a;=:  — 15  ;  .-.  completing  the  square 
&c.  a:=5,  or  3. 

But,  by  taking  the  negative  sign,  x-\-  -/(x^  — 9)  =  — 3a;4-6  ; 
.-.  by  transposing  and  squaring,  a:^  — 9  =  1 6a;2— 48a; -|- 36,  and 

by  transposition  and  division,  x^ — — a;=— 3  ;  completing  the 

o 

,     16       64         11  ,.       , 

squarmg,  a;2 — — a?+— =— — ;  .*.  takmg  the  r#ot  and  trans- 

O  tiO  liO 

posmg,  x= -*^ . 

17 
Ex.  4.  Given  a:*+-— a;^— 34a?=16,  to  find  the  values  of  x. 
2 

17 
By  transposition,  a;*-f— -a;^  =34a;  +  16  ;    completing  the 
2 


square,  x^-\'—x^-{-\-~xy=\-r-xy+^Ax'{-\Q\  .-.extracting 

the  root,  x^-{-—x=±\-—x-^Ay 

Let  the  positive  root  be  taken ;  then,  by  transposition,  x^-zzA; 
•.a;=2,  or  —2. 


QUADRATIC  EQUATIONS.  249 


17  1.7 

But  if  the  negative  value  be  taken,  x^-^—xz=  — —  a?  —  4 ; 

^17  ^  ,    ,     17     ,  289     289      ^      225 

•'•  ^^^"2-^= -4  ;  and  x^^-x  +  -~=-j-  -4=^g-;.-.ex. 

17         15 
tracting  the  root,  x-{-  —  =  ^—,  and  by  transposition,  ir=  —  8, 

or-f 

Ex.  5.  Given  4x'^—3x=-85,  to  find  the  values  of  x. 

Multiplying   by    16,  6Ax^—i8x=:l360,    and,    adding   the 
square  of  3,  64a;2_48a;_|_9  =  i369  ; 
/.extracting  the  square  root,  8a;  — 3=^37  ; 
by  transposition,  8a;  =  40,  or  —34,  .-.  a;=5,  or  — 4^. 

Ex.  6.  Given  6x-\ =44,  to  find  the  values  of  a:.    • 

X  • 

Multiplying  by  x,  6a;2+35  — 3a;=i44a;; 

.-.  by  transposition,  6x^—47x=  —35  ; 

47             35 
and  by  division,  x^ e~*= ^  »  therefore  completing  the 

O  D  *  " 

,      47       /47\*     2209       35     1369 
square,  x^-  -x+  (j^) ^=^^  "  -^=14?  '    •*•  ^^'^^^'^"^ 

47       .  37 

l2 


the  root,  x —  t^=  iTT'  and  x=7,  or  f 
1  -ii 


2x 3  3x 6 

Ex.  7.  Given  5x '■ — ~=2x-] — ,  to  find  the  values 

x—3  '       2     '       • 

of  a;. 

Multiplying  by  2a!— 6,  we  have  lOa;^  — 36a;4-6=4a;2— 12af 

+  3x2— 15a:+18; 

.'.  by  transposition,  3a;2 — 9a?=:12  ; 

and  by  division,  x^  —  3x:=4: 

.'.  completing  the  square,  a;^  — 3a;+f =4-f  |-=2^*, 

and  extracting  the  root,  a;— §=^1^ ;  ^ 

.-.  a?=:4,  or  —1.  ^ 

.  r.      n    r.-         ^         3a:— 10     ^  ,  6a;2— 40         ^    ,    ^ 
Ex.  8.  Given  3a;—  - — -—=2 +  -7^ r-,  to  find  the  va- 

9— 2a;  2a;— 1 

lues  of  X. 

Multiplying  by  2a;— 1, 

^  „     ^^       6a;2— 23a;-f  10       ,         .  ,  ^  o      .^ 

6a;2— 3a; — -— =  4a;— 2+6a;2— 40. 

9 — 2a; 

„     ,  6a;2— 23a; -I- 10 
or  7.+— -^^-—=42; 

•.•.63a;-14a;2-f6a;2--23a;4-10=:378-84ar; 
by  transposition,  124a?— 8a;2= 368, 


250  QUADRATIC  EQUATIONS. 

and  X — ^^ir=— 46  ;  .'.by  completing  the  square, 

2_21    ,961_961       .fi_g25 

""        2 '^'"^  16-16  16' 

31  15 

/.  extracting  the  root,  x =:^ — ;  • 

23 
and  therefore  x=z~,  or  4. 

Ex.  9.  Given  -y/a;^+y'a;3r=6-y/a;,  to  find  the  values  of  a. 

Dividing  by  \/x,  x^-\-x=6  : 
.*.  completing  the  square,  a;2-f-a;+^=6+J= V  » 
and  extracting  the  root,  a;+ J=z  j- J  ; 
.•.a:=:2^or  —3. 

n 

.  Ex.  10.  Given  a;" — 2ax'^  =bj  to  find  the  values  of  a?. 
Completing  the  square,  x"  —2ax^  -{•a^=a^-{-b  ; 

n 

.*•  extracting  the  root,  a;'  —a=^  •v/("^+^)» 
anda;«  =a±VK+*);  ••- cc=(a±^(a^-\-b)Y  . 

Ex.  11.  Given  a;2_2a:4.6y(x2— 2x+5)  =  ll,  to  find  the 
values  of  x. 

Adding  5  to  each  side  of  the  equation, 

(a;2— 2a;+5)+6v'(a^2_2a;4-5)  =  16  ; 
.'.  by  completing  the  square, 

(a;2-2x-l^5)-f-6V(a;2— 2a;+5)  +  9=25  ; 
and  extracting  the  root,  -v/(a;2— 2a;-|-5)  +  3  =  ±5  ; 
.-.  'v/(a;2— 2a?+5)=2,  or  -8  ; 
.-.  squaring  both  sides,  a;^— 2a?4-5=4,  or  64  ; 
-wAence  a;^— 2x-i-l=0,  or  60; 
and  extracting  the  root,  a;  — 1  =0,  or  rt  V^O  ; 

.-.  a;  =  l,  or  1±V'60. 

Sc^oL.  It  is  proper  to  observe,  that  the  equation,  a:^  — 2a: 
+  l,"s  two  equal  roots,  although  x  appears  to  have  only  one 
value  ;  but  it  is  because  x  is  twice  found  =1,  as  the  common 
method  of  resolution  shows  ;  for  we  have  xz=l±'\/0,  that  is 
to  say,  X  is  in  two  ways  =1. 

Ex.  12.  Given  x*+4x^+l2x^+16x  —  a,  to  find  the  values 
of  a;. 

Here  the  two  first  terms  of  the  square  root  of  the  left-hand 
member  (Art.  238),  is  found  to  be  x^-{-2x,  and  the  remainder 
is  8a;2+16a:,  which  can  be  readily  resolved  into  the  factors  8 
and  a;2-f2a;,  since  (8a;2+16a;)-r-(a:2-|-2a:)  gives  8  for  the  quo- 
tient. Consequently  the  proposed  equation  may  be  exhibited 
under  the  quadratic  form  (a:2-f-2a:)2-f  8(a;2-f  2a:)=a  ; 


QUADRATIC  EQUATIONS.  251 

/.by  completing  the  square,  {x'^-\-2xY-\-8{x'^-\-2x)-\-l6=a-{- 
16;  and  extracting  the  root,  x'^-\-2x-\-4:  =  :^y^(a-\-i6). 
Now  by  taking  the  positive  sign, 

a;2-|-2a:+4=:  +  y(a-f  16)  ; 
by  transposition,  x'^-i-2x=-—4-^-\/{a-\-l6) ; 
.-.  completing  the  square,  x'^-\r2x-i-l=z—3-\-'\/{a-\'l6)  ; 
and  extracting  the  root,  x-}-l  =  ^^{—3-{-  ^/a-\-l6)) ; 
.•.a:=-l±V(-3-j-V(«+16)). 
Agc^n,  by  taking  the  negative  sign, 

a:2+2a:+4  =  — V(«+16); 
...p-f2j:=— 4— V(arf-16);  and 
completing  the  square,  a:+2a;-|-l  =  — 3  —  -^{a-^-lS); 
.-.  extracting  the  root,  ac+l^iVl — 3  — y'(a*-|-16)).; 
and  x=  —  l±^{  —  3  —  -{/(a-^l6)). 
Ex.  13.  Given  3a;2—12a;H- 12  =  16—4,  to  find  the  values 

of  Of. 

By  transposition,  3a:2— 12a!:=:16  — 4  — 12  =  0 
and  by  division,  x^ — 4a3=0 
.-.by  completing  the  sqnare,  a;^— 4a;4-4=4 
and  extracting  the  root,  a;— 2  =  ;j::2 
•  .'.x=4,  or  0. 

Ex.  14.  Given  x^—4:X^-{-6x=4,  to  find  the  values  of  x. 
Multiplying  both  sides  by  x,  x!^—4x^+6x^—4x=0, 

.-.  {x^-2xf-\-2{x^—2x)  =  0, 
..a;2-2a?+l  =  dLl,anda:=liV±l  ; 
.•.the  three  roots  of  the  proposed  equation,  are  1,  1  +  y  — 1, 
and  l  —  -\/  —  1 .  The  other  value  of  x,  which  is  equal  to  1  —  1 , 
or  0,  belongs  to  the  equation  {x'^—2xY-{-2{x'^—2x)  —  0  ;  hence 
there  are  four  roots,  or  four  values  of  x,  which  will  satisfy  this 
last  equation. 

Ex.  15.  Given  27a;2_  |li+.ll=?5?_i.^_|_5  to  find  the 
dx^       3        3a;      3x^ 

values  of  a:. 

Multiplying  every  term  by  3,  ^ 

X^  X  x^ 

1        R4-1       2^2 
.-.by  transposition,  Q\x'^-\-\l-\ — ^=— ^ — I — 2"^^^' 

Adding  unity  to  each  side,  in  order  to  complete  the  square  ; 

X^         X^  X 

1           29 
and  extracting  the  root,  9a; H — =zl=( 1-4). 


252  QUADRATIC  EQUATIONS. 

Let  the  positive  value  be  taken ;  then  by  transposition,  9a? 

28 
— 4= — ,  and  .-.  9x^—4:X=28  ;  by  completing  the  square,  &c., 

14 
we  shall  have  x=z2,  or  — —.     But  if  the  negative  value  be 

taken,  9a!'^-f-4a;=— 30  and  completing^he  square,  &c.,  a; = 
— 2^-v/(-266) 

Ex.  16.  Given  3a;2-|-2a;— 9=76,  to  find  the  walues  of  a:. 

Ans..=5,or-n. 

T<       !«#-.•'    8-— a;     2a;— 11      a:— 2  ^    ■,     -,        '  i 

Ex.  17.  Given—- -— =— -^   to   find   the   values 

2  x—3  6 

of  a?.  Ans.  a;=6,  or,  ^. 

„       ,o    r.-        3a:+4     30-2a:      7a;  — 14       ^    ,  , 

Lx.  18.  Given  — r— = — r^r — ,  to  find  the  values 

5  J?— 6  10 

of  a?.  Ans.  a;=36,  or  12. 

a!3_i0a:2+l 
Ex.  19.  Given  — - — — — p— -=a;— 3,  to  find  the  values  of  x. 
a;2— 6a;+8 

Ans.  a;=l,  or  —28. 

Ex.  20.  Given-/(a;+5)X'v/(a;+12)  =  12,  to  find  the  values 

of  a;  Ans.  a;=4,  or  —21. 

Ex.  21.  Given  2a;2+3a:— 5V'(2a;2+3a;-|-9)  +  3=0,  to  find 

9        — 3iV— 55 

the  values  of  x.  Ans.  a:=3,  or  — -,  or f- 

2  4 

Ex.  22.  Given  9a;-h  V(16a;2  +  36a;3):=15a;2— 4,  to  find  the 

,  .  .41*       9i-v/481 

values  of  x.  Ans.  a?=-,  or  — -  ;  or  ^ — 

3  3  50 

49^2     48  6 

Ex.  23.  Given  +-^—49=9+-,  to    find   the  values 

4        a;2  X 

8        — 3J::'/93 
ofar.  Ans.  a;=2,  or — ^^  or — : — — . 

Ex.  24.  Given  a;^— 2^3+ a; =132,  to  find  the  values  of  a;. 

Ans.  a;=4,  or  — 3,  or ^ . 

6  3 

Ex.  25.  Given  a;*+a;*— 756,  to  find  the  values  of  x. 

Ans.  a;=243,  or  (—28^. 

3 

Ex.  26.  Given  a:^— a;^=56,  to  find  the  values  of  a;. 

2 
Ans.  a;=4,or  (— 7)^. 


QUADRATIC  EQUATIONS.  253 

Ex.  27.  Given  07+5= -/(ac-i- 5) +6,  to  find  the  values  of  ae. 

Ans.  x=4:,  or  — 1. 

Ex.  28.  Given  x+l6—7 ■^(x-{-l6)  =  l0—4^{x-i-l6),  to 

find  the  values  of  a:.  Ans.  x=9,  or  —12. 

Ex.  29.  Given  x-\-4-\ =13,  to  find  the  values  of  od* 

X 

Ans.  a;=4,  or  — 2. 

Ex,  30.  Given  14+4a;-  ^=3a:+^^-,   to   find   the 
x—1  3 

values  of  x.  Ans.  «=28,  or  9. 

Ex.  31.  Given  -^ -=     J     — 1,  to  find  the  va- 

3  X — 3         9 

lues  oix.  Ans.  a:=21,  or  5. 

Ex.  32.  Given  20^  +  18-  ^f  ^!f  =27-  ^^"^"y, to  find 
4a;+7  2a; — 3 

the  values  of  x.  Ans.  a; =8,  or  5. 

Ex.  33.  Given  — ^-^ -^  =a;^-fj?-f  8,  to  find  the  values 

x^'\-x—Q 

of  a-.  Ans.  a:=4,  or  — 4|. 

Ex.  34.  Given  <v/(4a?+5)X'/(7a:-fl)  =30,  to  find  the  va- 
lues of  a:.  Ans.  a; =5,  or  —  6^-. 

Ex.  35.  Given  — 1— -^=-^  to  find  the  values  of  x. 

X         a?4-12     15 

Ans.  a:=3,  or  — 15. 
Ex.  36.  Given  a;^-|-7a;^=44,  to  find  the  values  of  x. 

3 

Ans.  a;=-[-8,  or  ±(  —  11^- 

Ex.  37.  Given  4a:^  +  a;^  =  39,  to  find  the  values  of  a;. 

Ans.  a:=729,  or  (—V 

Ex.  38.  Given  3a;6+42a:3:=3321,  to  find  the  values  of  x. 

Ans.  a;=3,or  — ?/41. 
8  17 

Ex.  39.  Given  —4-2=-^,  to  find  the  values  of  x. 

X  -35 

X 

Ans.  a;=4,  or  J?/ 2. 
Ex.  40.  Given  a;2+ll-|-y(a;2  +  ii)^42,  to  find  the  va- 
lues of  X.  ,  Ans.  x=:i^5,  or  -t  -y/SS. 
Ex.  41.  Given  a;2_12a;+50  =  0,  to  find  the  values  of  a;. 

Ans.  a:=6±V'(  — 14). 
Ex.  42.  Given  3a;— ^a;2  =  10,  to  find  the  values  of  x.  ' 

Ans.  a;=64:-v/— 4. 
23 


254  QUADRATIC  EQUATIONS. 

Ex.  43.  Given  x^— 2x^  =  48,  to  find  the  values  of  a?. 

Ans.  x  =  2,  or  3/  —6. 

Ex.  44.  Given  a:*-f  2x2— 7a2_8a:=:-12,  to  find  the  va- 
lues of  X.  Ans.  2,  or  —3,  or  1,  or  —2. 

Ex.  45.  Given  x*— 10a-^+35»2_50a+24=iO,  to  find  the 
values  of  x.  Ans.  x=:l,  2,  3,  or  4. 

Ex.  46.  Given  a;^  — 8x2+190'— 12  =  0,  to  find  the  values 
of  ar.  Ans,  a;=l,  3,  or  4 

Ex.  47.  Given ^-=z '■,  to  find  ihe  values  of  x. 

x—yx         4 

Ans,  a?=4,  or  1,  or  liJV"""^* 

Ex.  48.  Given  4x4 4- ^=4*3 +  33,  to-find  the  values  of  a:. 
2 

Aas,  x=2,  or  —  -  ;  or  — -, 

2  4 

§  II.    SOLUTION    G-F   AI>fECTE©    QUADRATIC    EQUATIONS,  IN- 
VOLVING TWO  UNKNOWN  QUANTITIES. 

360.  When  there  are  two  equations  containing  two  un- 
known quantities,  a  single  equation,  involving  only  one  of  the 
unknown  quantities,  may  sometimes  be  obtained,  by  the  rules 
laid  down  for  the  solution  of  simple  equations  ;  from  which 
equation  the  values  of  the  unknown  quantity  may  be  found, 
as  in  the  preceding  Section.  Whence,  by  substitution,  the 
values  of  the  other  may  also  be  determined.  In  many  cases, 
however,  it  may  be  more  convenient  to  solve  one  or  both  of 
the  equations  first ;  that  is,  to  find  the  values  of  one  of  the  un- 
known quantities,  in  terms  of  the  other  and  known  quantities, 
as  before  ;  when  the  rules  for  eliminating  unknown  quantities, 
(^  I.  Chap.  IV),  may  be  more  easily  applied. 

The  solution  will  sometimes  be  rendered  more  simple  by 
particular  artifices ;  the  proper  application  of  which  shall  be 
illustrated  in  the  following  examples. 

i'    9  Vt^  ^     9^~^o'  f  to  find  the  values  of  x  and  y. 
&na  x^+3xi/—y^=23,S  ^ 

From  the  1st  equation  x-=l  —  2y  ; 
...j:2_49_28y+4y2; 

Substituting  these  values  for  x  and  a;2  in  the  2d  equation, 
then  49— 28y+4y2+21y— 6y2_y2_23, 

or  3y2+7y:::349-23=26. 

36y2+84y  +  49==312  +  49  =  361  ; 

.'.extracting  the  square  root,  6y+7=19, 

and  6^=19—7=12;  y=3, 

and  a?=7— 2y=7— 4  =  .^ 


QUADRATIC  EQUATIONS.  255 

Ex.  2.  Given  4ry=96— aj^,  and  :c+y=6,  to  find  the  va- 
lues of  X  and  y. 

From  the  first  equation  a;y  4-4ary+4=100, 
and  extracting  the  root,  a:y-|-2=:  ±10  ; 
.-.  a?y=8,  or  —12. 
Now  squaring  the  second  equation, 

«24.2j:y-f  y2  =  36  ; 
but       4a:y         =32,  or  —48. 

.*.  by  subtraction,  *2— 2a;y4-y^  — 4,  or  84  ; 
and  extracting  the  root,  a:— ^==±2,  or ^^ -1/84  ; 
but  X'\-y=^     6; 


.-.by  addition,  2a:=8,  or  4,  or  6rt  ■v/84  ; 

whence,  «=:4,  or  2,  or  3  ^^-^^21 ; 

and  by  subtraction,  2y=4,  or  8,  or  6^-^/84  ; 

.•.y=2,  or  4,  or  3=PV21. 

Ex.  3.  Given  ac2-^a;-fy=18—y^  and  a:y= 6,  to  find  the  va- 
lues of  X  and  y. 

By  transposition,  x^-\-y'^-\-x-\-y—\^\ 
and  from  the  second  equation,  2a;y  1=12  ; 


.-.  by  addition,  a:2-f-2a:y-f  y2-|_a?-|-y=30  ; 
and  completing  the  square, 

(»:+#+(*+y)+-l=30+i=^ ; 

.•.extracting  the  root,  .-K-f-y+i^iy* 

and  a;4-y=:5,  or  — 6  ; 

whence,  from  the  first  equation,  a;2-|-y''i=13,  or  24; 

but  2a:y=12; 

.-.by  subtraction,  a:^— 2a7y  +  y2:=il,  or  12 

.*.  extracting  the  root,  x — y  =  drl,or  ::t2y'3 

Now  aj+y  =  5,  or  — 6 

.-.  by  addition,  2r=z6,  or  4,  or  — 6±2-/3 

.•.a;=3,  or  2,  or  — SJ^-y/S 

and  by  subtraction,  2y=4,  or  6,  or  — 6  =F 2 1/3 

.•.y=2,  or  3,  or  —  3=F  \/2. 

Ex.  4.  Given*— 2'v/a:y4-y  — -v/^-fVy^^' ^"^  \f^-\-y/y 
=r5,  to  find  the  values  of  x  and  y. 

Completing  the  square  in  the  first  equation, 

and  extracting  the  root,  y/ x —  -/y — \z=:i-^\  ; 


256  QUADRATIC  EQUATIONS. 

.-.  ^/x—  -y/y,  =1  or  0 
but  from  the  second  equation,  ^/x-^  ^Jy~^  > 


/.  by  addition,  2'/a:=z6,  or  5, 

5  25 

and  ^aj=3,  or  -,  .-.icnrQ,  or  — . 
2  4 

25 
By  subtraction,  2-v/y=4,  or  5  ;  .•.y=:4,  or  — . 

2.    3  \        \ 

Ex.  5.  Given  a;^  y'^  =2y2,  and  Sa-^  — y^  =  14,  to  find  the  va- 
lues of  X  and  y. 

2  1  2  1. 

From  the  1st  equation,  a?3=2y2  ;  and  :.\x^z=iy'^  ;   substi- 
tuting this  value  in  the  second  equation, 

Sa;^— Aa;3  =  l4  ;   and  .-.  16a;^— a:^:=28  ; 
2.  X 

or,  by  changing  the  signs,  x^  —  \%x^=i — 28  ; 
2.  X 

completing  the  square,  a?3  —  16a;3  + 64=36  ; 

and  extracting  the  root,  x^^%=l^^  ; 
J, 
.•.a;3  =  14,  or  2,  and  ac=i2744,  or  8. 

JL  2  \  \_ 

Ex.  6.  Given  a;^+y^  =  3a:,  and  a;^+y^=:a?,  to  find  the  va- 
lues of  X  and  y. 

11.  2 

By  squaring  the  second  equation,  x-\-2x^y^  ■\-y'^  =0^- ; 

3  2 

but  a; 2  -fy3=:3a;; 

3  11 

.-.by  subtraction,  x-^o^ -\-2x^y^ ^x^ —"^x  ; 

\  1 

but  from  the  second  equation,  y^-=ix~3^\ 
Let  this  value  be  substituted  in  the  preceding  equation  ; 

then  X — o[-^-\-2x^—2x=x^ — 3j;  ; 

3 

.*.  by  transposition,  2x=x^—x^  ; 

and  dividing  by  x,  2=x—x^; 
1 
completing  the  square,  a;— ;c^4-J=:2-f -1=1- ; 

and  extracting  the  root  a:^— J=-|-J  ; 

,-.  x'^—2,  or  —  1  ;  and  a;=4,  or  1. 

By  taking  the  former  value  of  x,  we  have  y^=zx — x^ 

=4  -2=2;  .'.y  =  8. 

11 
and  by  taking  the  latter  value,  y^z^a;— a;^=l-|-l=2, 

t8inceaj^=:  — 1,  — a;i=-i-l);  /.  y=8 


QUADRATIC  EQUATIONS.  257 

Ex.7.  Given  y2_64=8a;2y,andy— 4=2y^a:^,  tofind  the 
values  of  x  and  y. 

From  the  first  equation,  y^— 8x^2/ =64  ; 

completing  the  square,  y^ — 8x^y-{-l6x=16x-{-64  ; 

extracting  the  root,  y— •4a;2  =  -|-4'v/(a:+4) ;  • 

and  .•.y=:4a;2_j_4-y/(a;-|-4). 

Also,  from  the  second  equation,  y—2y^x^=4; 

.'.completing  the  square,  &;c.,  y'^=zx^di  '\/{x-\-4) ; 

multiplying  by  4,  4y2=a;2_|^4y(a:-f  4) ; 

.-.  y=4y^,  and  y=16. 

But,  from  the  second  equation,  x^=- — j=—-=-  ; 

.-.  by  involution,  a;=-, 
4 

361.  When  the  equations  are  homogeneous,  that  is,  when 
flc^j  y2,  or  a:y,  is  found  in  every  term  of  the  two  equations,  they 
assume  the  form  of 

ax"^  -\-  bxy  -\-  cy"^ = d, 

a'x'^-{-h'xy-\-c'y^=d' ;  and  their  solution  may  be  effected 
in  the  following  manner  : 

Assume  xz^vy,  then  x^z^zv^y"^ ;  by  substituting  these  values 
for  x"^  and  x  in  both  equations,  we  have 

avY+bvy^-^cy^=d  I   /.y2^_____     .     .     .     (1), 

a'vY+b^vy^  +  cY==d^;    r.f=-^^-^,    ..     .     .     (2). 

^  d  d^ 

Hence 


av^-{-bv-i-c     a'v'^-\-b'v-{-c' 

.'.(a'd—ad')v'^-\-{b'd—bd')v=cd'  —  c'd;  which  is  a  quadratic 
equation,  from  whence  the  value  of  v  may  be  determined. 
Having  the  value  of  v,  the  value  of  y  may  be  found  from  ei- 
ther-of  the  equations  (1)  or  (2)  ;  and  then  the  value  of  x,  from 
the  equation  x  =  vy. 

Ex.  8.  Given  2a;2+3a;y+y2=:20,  and  5a;24-4y2=41,  to  find 
the  values  of  x  and  y. 

Let  x=vy,  then  2v^y^'i-3vy^-\-y^=20  ; 
23* 


258  QUADRATIC  EQUATIONS. 

20 

•'•y  =A  2  L  ^  >  Hence  .  ^— ;:  9  ,  .>  o^  6t;2— 41t;= 

-13; 

.•.by  division,  completing  the  square,  &c.  u=^2^  or  ^. 

•  41  369 

Let  t;=i    then  y2^___=___^9  .  ...y^g^  or  -3, 

and  xz=zvy=z\,  or  —1. 

A      •     1  13      ,  ,      ,164         ,  .    13    ,164 

Again,  let  «=—;  theny=±V— ,  and  a;=±— -/-— . 

Consequently  there  are  four  values,  both  of  x  and  y,  which 
satisfy  the  proposed  equations. 

362.  When  the  unknown  quantities  in  each  equation  are 
similarly  involved,  the  operation  may  sometimes  be  shortened, 
by  substituting  for  the  unknown  quantities  the  sura  and  differ- 
ence of  two  others. 

Ex.  9.  Given-+^-18,)       .    ,  ,        .  .         , 

y     X  >  to  find  the  values  of  x  and  y. 

and  a;+y=12,  ) 

Assume  x=zz-{-v^  and  y=.z—v  ;  .•.a;+y=2;?=12  ; 

or  0=6;  .-.  a;=6-j-t;,  and  y=6 — v. 

Also,  since }-^=18,  x^-\-y^=^\Sxy  ; 

y       x  J  J 

.-.  (6  +  vV  +  {6-t^)3  =  18(64-«)  X  (6-t;)  ; 
or  432  +  36^2-648  — 18i;2  ; 
and  by  transposition,  54i;2=216  ; 

...v2=4  .  and  t;=±2  ;  .-.  a?=6i2  =  8,  or  4  ; 
and  y=6±2=:4,  or  8. 

363.  In  all  quadratics  of  this  kind,  in  which  x  may  be 
changed  for  y,  and  y  for  a;,  in  the  original  equations,  without 
altering  their  form,  the  two  values  of  one  of  the  quantities  may 
be  taken  for  the  values  of  the  two  quantities  sought. 

Ex.  10.  Given  x-\-y=2a,  and  x'^^y^—h,  to  find  the  values 
of  X  and  y. 

Let  x—yr=i2z  ;  then  a;=:a+«,  and  y=a — z  ; 
.-.by  substitution,  (a+;?)5+(cr— 0)-^=J,  or,  by  involution  and 
addition,  2a^^-1^a^z''-^-\^az''  =  h  ; 

.'.x^a^^Jl-a^±^/^^^)\  and  y=.a^./\-a^^ 

^("lor"- 


QUADRATIC  EQUATIONS.  259 ' 

Now,  let  x-\-y  =  6,  and  a;5-|-y5=1056  ;  then  by  substituting 
3  for  a,  and  1056  for  b,  in  the  formula  of  roots,  the  values  of 
a?andy  will  be  found  ;  that  is,  a?  — 3  J::!,  or  Si-yZ  —  lO  ;  and 
y=:3=f  1,  or  3=p-v/  — 19-  Or,  by  substituting  the  above  va- 
lues of  a  and  b  in  the  equation  lOaz"^ '-20a^z^-\-2a^  —  b,  it  be- 
comes 30;2'*+540z+486  =  1056  ;  from  which  the  values  of  z 
may  be  found  ;  whence,  by  substitution,  the  values  of  x  andy 
will  be  determined,  as  before. 

Ex.  11.  Given  a;+4y  =  i4i  and  y24-4a?=2y+ll,  to  find 
the  values  of  x  and  y. 

Ans.  a;=— 46,  or  2  ;  and  y=15,  or  3. 

Ex.  12.  Given  2a;-f-3y=118,  and  5a;2— 7y2=:4333,  to  find 

the  values  of  x  and  y.    ' 

3899         _,         ,^       3268 

Ans.  a;=35,  or :  and  y=16,  or-——. 

17  -^  17 

Ex.  13.  Given  x^+4i/^=256—4x7/,  and  3y2-.a;2  =  39,  to 
find  the  values  of  x  and  y. 

Ans.  a?=i6,  or  il02  ;  and  y=:i5,  or  ±59. 
Ex.  14.  Given  a^-f  y''=2a'',  and  xy=c'^,  to  find  the  values 
of  X  and  y. 

]_  • 

a;=[a''-l-V'(o«"--c"')]''; 

Ans.  <f  ■■_  g" 


[a"±^(a"'-c'")]'' 

Ex.  15.  Given  a:2+2a;y+y2+2a:  =  120  — 2y,  and  a:y—y2=: 
8,  to  find  the  values  of  x  and  y. 

Ans.  x—6,  or  9,  or  — 9^'v/5  ;  andy=4,  or  1,  or  —3  j^ 
V5. 

Ex.  16.  Given  x^-^y^ — x—y  =  78j  and  a;y+a: 4-y=39,  to 
find  the  vabies  of  x  and  y. 

Ans.  a;— 9,  or  3;  or  — 6JJ::^y'  — 39;  and  y=3,  or  9,  or 

Ex.  17.  Given  _„4— =-:r-,  ^ ...  gn^  j^^  values  of  a;  and  y. 


\2+y-    9'StO 

and  a?— y=2,  ^ 


17  3 

Ans.  x=z5,  or  —  ;  and  y=3,  or  —  — . 

Ex.   18.  Given  a;*— 2a;2y-l-y2=49,  >  to  find  the  values  of  a; 
and  a;* — 2x^y^'{-y'^—x^+y'^=20,  >  and  y. 

Ans.  a;=-t3,  or  iV^,  or  :kWi^^:^^V^)  i 
andy=2,  or  -1,  or  i(l±3  ^5).* 

*  There  are  four  other  values,  both  of*  and  y,  which  are  all  imaginary. 


%0  QUADRATIC  EQUATIONS 

i  1 

Ex.  19.  Given  4— a;^=:3— y,  and  4— aj^ry— y^jtofindthe 
values  of  x  and  f.  Ans.  x=4,  or  i  ;  and  y  =  l,  or  2^. 

3  i 

Ex.  20.  Given  a: 2  + a:— 4x2  =y2 4- y 4- 2,  and  xi/=y^-{-3y, 
to  find  the  values  of  x  and  y. 

Ans.  a:=-4,  or  1  ;  and  y  =  l,  or  — 2. 

Ex.  21.  Given  a:^+a:y=56,  and  a:y  +  2y2=60,  to  find  the 

values  of  x  and  y.  Ans.  x=  dL4'v/2,  or  4=  1 4  ; 

'      and  yr=i3^2,  or  ±10. 

Ex.  22.  Given  a;— y  =  15,  and  xyz=.2y^,  to  find  the  values 

of  a:  and  y.  Ans.  a;=18,  or  12j  ;  and  y=r3,  or  — 2^. 

Ex.  23.  Given  10a;-j-yr:z3a;y,  and  9y  — 9ar=18,  to  find  the 

values  of  a;  and  y.  Ans.  a:=2,  or  —\  ;  and  y  =  4,  or  J. 

Ex.  24.  Given  .x+y  :  x—y  :  :  13  :  5,  >  to  find  the  values 

and  y''*+a:r^25,  J  of  a;  and  y. 

Ans.  a?=9,  or  —14^^;  and  y=4,  or  — 6^^. 

Ex.  25.  Given  a?2y*— 7a'y2=rl710,  and  a^y—y  =  12,  to  find 

the  values  of  x  and  y. 

—  19 
Ans.  a:=5,  or  J,  or  ■        ^     — -  ;  and  y=3,  or  —15,  or  — 

6iV-2.     • 

Ex.  26.   Given  a:y-|-a:y2  =  12,  and  a;+a:y^=18,  to  find  the 
values  of  a?  and  y.  Ans.  a;=2^  or  16  ;  and  y=:2,  or  ^. 

Ex.  27.  Given  a:4-y  +  'v/(a'+y)=6,  and  a;2+y2.^10,  tofind 
the  values  of  x  and  y. 

Ans.  iK— 3,  or   1  ;  or  4^^\^/ —Ql  ;  and  y=l,  or,  3,  or 

4iTiV-6i. 

Ex.  28.  Given  a;2-|-4^(a:2+3y+5)  =  55— 3y,  and  6a;— 7y 

.=  16,  to  find  the  values  of  x  and  y. 

-53          — 9±V5072 
a;=5,  or  -y-  ;  or ^ . 

^^^'  ^        „              430          -166^6^5072 
y=2,or-— ;or . 

Ex.  29.  Given  a;24-2a:3y=441~a:y,  and  a:y=34-a:,  to  find 
the  values  of  a;  and  y. 

.        Ja:=:3,or  -7;  or  -2^^-17, 
A"«-^y=2,or|;  or|T|V-17. 
Ex.  30.  Given(a:H-y)^  — 3yr=28H-3ar,  and2a;y+3a:=35,to 
find  the  values  of  x  and  y. 

<  a;=5,  or  5,  or  -^^J^\^{-^2b5\ 
^"'-  \  y=2,or  I,  or  -U  T  V'i(-255.) 
Ex.  31.  Given  a:24-3a;+y=73— 2a:y,  andy2+3y4-ar=44, 
to  find  the  values  of  x  and  y- 

.        5a:=4,  or  16;  or  —  12T'v/58, 
^^^'  \  y=5,  or  -7  ;  or  -li  -v/SS. 


SOLUTION  OF  PROBLEMS,  &c.  261 

Ex.  32.  Given^+^=136J— 2a:y,anda;+y=10,  tofind 

y     ^ 

the  values  of  x  and  y. 

,         (a:z=6,or4;  ox  5^b^(-\\,) 
''^"^-  ^y=4,or6;  or  5  ={=5 ^ -(H)- 
Ex.  33.  Given  y*-432  =  12a:y2,  and  y^^-l^  +  ^xy,  to  find 
the  values  of  x  and  y. 

Ans.  x=2,  or  3  ;  and  y=:6,  or-v/(21)+3. 


CHAPTER  XL 


THE  SOLUTION  OF  PROBLEMS, 

PRODUCING  aUADRATIC  EaUATIONS. 

§  I.    SOLUTION  OF  PROBLEMS  PRODUCING  QUADRATIC  EQUA- 
TIONS, INVOLVING  ONLY  ONE  UNKNOWN   QUANTITY. 

364.  It  may  be  observed,  that,  in  the  solution  of  problems 
which  involve  quadratic  equations,  we  sometimes  deduce, 
from  the  algebraical  process,  answers  which  do  not  correspond 
with  the  conditions.  The  reason  seems  to  be,  that  the  alge- 
braical expression  is  more  general  than  the  common  language, 
and  the  equation,  which  is  a  proper  representation  of  the  con- 
ditions, will  express  other  conditions,  and  answer  other  suppo- 
sitions. 

Prob.  1.  A  person  bought  a  certain  number  of  oxen  for  80 
guineas,  and  if  he  had  bought  four  more  for  the  same  sum, 
they  would  have  cost  a  guinea  a  piece  less.  Required  the 
number  of  oxen  and  price  of  each. 

Let  x-=.  the  number  ;  then  —zzz  the  price  of  each  ; 

x 

=r — ^  — 1,  by  the  problem, 


a:+4       X 

and  by  reduction,  a;2+4a:==320  ; 

.-.  a?2-}-4a;+4:=;324,  and  a:+2  =  il8; 

.•.a;=16,  or  —20. 


262  SOLUTION  OF  PROBLEMS 

A    ^80     80  .  ^        •        r       ^ 

And  — =——5  guineas,  the  price  of  each. 

The  negative  value  (—20)  of  x,  will  not  answer  the  con- 
dition of  the  problem. 

Prob.  2.  There  are  two  numbers  whose  difference  is  9,  and 

their  sum  multiplied  by  the  greater  produces  266.     What  are 

those  numbers  ? 

Let  xzzz  the  greater;  .-.aj— y=  the  less. 

9        2fifi 
and  x.{2x  —  9)=266  ;  .-.  x^—  -.x=~~.  ^ 

w       v  ,  9    •    47 

completing  the  square,  &c.  x — -=:  db-r  » 

.*.  a:r=:14,  or  —  9J;  and  a? — 9  =  5,  or  —  18J. 
Here  both  values  answer  the  conditions  of  the  problem. 

Prob.  3.  A  set  out  from  C  towards  D,  and  travelled  7  miles 
a  day.  After  he  had  gone  32  miles,  B  set  out  from  D  to- 
wards 0,  and  went  every  day  one-nineteenth  of  the  whole 
journey ;  and  after  he  had  travelled  as  many  days  as  he  went 
miles  in  one  day,  he  met  A.  Required  the  distance  of  the  places 
C  and  D. 

Suppose  the  distance  was  x  miles. 

.*.  Tq=  the  number  of  miles  B  travelled  per  day ;  and  also 

=  the  number  of  days  he  travelled  before  he  met  A. 

a;2                7x 
.-. l-32-j =a;: 

by  transposition  and  completing  the  square, 

extracting  the  root,  ^3—6=^2; 
1  y 

.-.  Y3=8,  or  4  ;  and  a!:=152,  or  76,  both  which  values  an- 
swer the  conditions  of  the  problem.  The  distance  therefore 
of  C  from  D  was  152,  or  76  miles. 

Prob.  4.  To  divide  the  number  30  into  two  such  parts,  that 
their  product  may  be  equal  to  eight  times  their  difference. 

Let  a;=:  the  lesser  part ;  .-.30— x=:  the  greater  part,  and 
30 — x—x,  or  30—20!;=  their  difference. 

Hence,  by  the  problem,  a(30  — a;)  =  8(30— 2a;),  or  30a:— «2 
=240- 16a;;  .•.a;^— 46a;=— 240. 


PRODUCING  QUADRATIC  EQUATIONS.     263 

.-.  completing  the  square,  rr2__46a?+ 529=289  ; 

.-.  a:  — 23^17=40,  or  6=  lesser  part; 

and  30  — a;z=:30— 6=r24=  ^rea^er  part. 

In  tliis  case,  the  solution  of  the  equation  gives  40  and   6 

for  the  lesser  part.     Now  as  40  cannot  possibly  be  a  part  of 

30,  we  take  6   for  the  lesser  part,  which  gives  24  for   the 

greater  part ;  and   the  two  numbers,  24  and  6,  answer   the 

conditions  required. 

Prob.  5.  Some  bees  had  alighted  upon  a  tree  ;  atone  flight 
the  square  root  of  half  of  them  went  away  ;  at  another  eight- 
ninths  of  them  ;  two  bees  then  remained.  How  many  then 
alighted  on  the  tree  ? 

16x2 
Let  2jc2=  the  number  of  bees  ;  x-\ — - — |-2=2a;2, 

or  9x-f  16a;24-18  =  18a;2;   .-.  2a;2— 9a:=18  ; 

Multiplying  by  8,  16^2  — 72a;  =  144  ; 

adding  81  to  both  sides,  16a?2_72a;+81  =225  ; 

.-.  4a;=9i  15=24,  or  —6  ;  and  j;=6,  or  —  1^. 

.-.  2a;2  =  72,  or4f 
But  the  negative  value  — 1^  of  a?,  is  excluded  by   the  na- 
ture of  the  problem  ;  therefore,  72=  number  of  bees. 

365.  If,  in  a  problem  proposed  to  be  solved,  there  are  two 
quantities  sought,  whose  sum,  or  difference,  is  equal  to  a 
given  quantity,  for  instance,  2a  ;  let  half  their  difference,  or 
half  their  sum,  be  denoted  by  x  ;  then  a:-}-«  will  represent 
the  greater,  and  x—a  the  lesser,  (Art.  102).  According  to 
this  method  of  notation,  the  calculation  will  be  greatly  abridg- 
ed, and  the  solution  of  the  problem  will  often  be  rendered  very 
simple. 

Prob.  6.  The  sum  of  two  numbers  is  6,  and  the  sum  of 
their  4th  power  is  272.     What  are  the  numbers  1 

Let  a?=  half  the  difference  of  the  two  numbers  ;  then  3  + 
a:=  the  greater  number,  and  3  — a:=  the  lesser. 

.-.  by  the  problem,  (3  +  a:)++(3-a;)4  =  272, 
or  162  +  108a;2-|-2a;*=272  ;  from  which,  by  transposition  and 
division,  a!*+54a;2=55  : 

.-.  completing  the  square,  a;*+54a;2 4-729  =  784, 

and  extracting  the  root,  a;2-f  27=  Jr28  ; 

.-.  a;2==— 27±28,  and  a:=il,  or  J- V-55. 

Now,  by  taking  the  positive  value,  +1,  for  x,  (since  in  this 
case,  it  is  the  only  value  of  x  which  will  answer  the  problem) ; 
we  shall  have  3+1-4=  the  greater,  and  3  —  1=2=  the 
lesser. 


264  SOLUTION  OF  PROBLEMS 

Prob.  7.  To  divide  the  number  56  into  two  such  parts,  thai 
their  product  shall  be  640.  Ans.  40,  and  16. 

Prob.  8.  There  are  two  numbers  whose  difference  is  7, 
and  half  their  product  plus  30,  is  equal  to  the  square  of  the 
lesser  number.     What  are  the  numbers  ? 

Ans.  12,  and  19. 

Prob.  9,  A  and  B  set  out  at  the  same  time  to  a  place  at  the 
distance  of  150  miles.  A  travelled  3  miles  an  hour  faster  than 
B,  and  arrives  at  his  journey's  end  8  hours  and  20  minutes 
before  him.     At  what  rate  did  each  person  travel  per  hour  1 

Ans.  A  9,  and  B  6  miles  an  hour. 

Prob.  10.  The  difference  of  two  numbers  is  6  ;  and  if  47 
be  added  to  twice  the  square  of  the  lesser,  it  will  be  equal  to  the 
square  of  the  greater.     What  are  the  numbers  ? 

Ans.  17,  and  11. 

Prob.  11.  There  are  two  numbers  whose  product  is  120, 
if  2  be  added  to  the  lesser,  and  3  subtracted  from  the  greater,, 
the  product  of  the  sum  and  remainder  will  also  be  120.  What 
are  the  numbers?  Ans.  15,  and  8. 

Prob.  12.  A  person  bought  a  certain  number  of  sheep  for 
120Z.  If  there  had  have  been  8  more,  each  would  have  cost 
him  ten  shillings  less.     How  many  sheep  were  there  ? 

Ans.  40. 

Prob.  13.  A  Merchant  sold  a  quantity  of  brandy  for  39/. 
and  gained  as  much  per  cent  as  the  brandy  cost  him.  What 
was  the  price  of  the  brandy  ?  Ans.  30/. 

Prob.  14.  Two  partners,  A  and  B,  gained  18/.  by  trade. 
A's  money  was  in  trade  12  months,  and  he  received  for  his 
principal  and  gain  26/.  Also,  B's  money,  which  was  30/.  was 
in  trade  16  months.     What  money  did  A  put  into  trade  ? 

Ans.  20/. 

Prob.  15.  A  and  B  set  out  from  two  towns  which  were  at 
the  distance  of  247  miles,  and  travelled  the  direct  road  till 
they  met.  A  went  9  miles  a  day  ;  and  the  number  of  days, 
at  the  end  of  which  they  met,  was  greater  by  3  than  the 
number  of  miles  which  B  went  in  a  day.  How  many  miles 
did  each  go  ? 

Ans.  A  117,  and  B  130  miles. 

Prob.  16.  A  man  playing  at  hazard  won  at  the  first  throw, 
as  much  money  as  he  had  in  his  pocket  ;  at  the  second  throw, 
he  won  5  shillings  more  than  the  square  root  of  what  he  then 
had  ;  at  the  third  throw,  he  won  the  square  of  all  he  then  had  ; 
and  then  he  had  112/.  l^y.     What  had  he  at  first  ? 

Ans.  18  shillings. 


OK  To- 

;;  UNiVERsr: 

PRODUCING  QUADRATIC  EQUATir^^^^'^'"*'^' 

Prob.  17.  If  the  square  of  a  certain  number  be  taken  from 
40,  and  the  square  root  of  this  difference  be  increased  by  10, 
and  the  sum  multiplied  by  2,  and  the  product  divided  by  the 
number  itself,  the  quotient  will  be  4.     Required  the  number. 

Ans.  6. 

Prob.  18.  There  is  a  field  in  the  form  of  a  rectangular 
parallelogram,  whose  length  exceeds  the  breadth  by  16  yards  ; 
and  it  contains  960  square  yards.  Required  the  length  and 
breadth.  Ans.  40  and  24  yards. 

Prob.  19.  A  person  being  asked  his  age,  answered,  if  you 
add  the  square  root  of  it  to  half  of  it,  and  subtract  12,  there 
will  remain  nothing.     Required  his  age.  Ans.  16. 

Prob.  20.  To  find  a  number  from  the  cube  of  which,  if 
19  be  subtracted,  and  the  remainder  multiplied  by  that  cube, 
the  product  shall  be  216.  Ans.  3,  or  —2. 

Prob.  21.  To  find  a  number,  from  the  double  of  which  if 
you  subtract  12,  the  square  of  the  remainder,  minus  1,  will  be 
9  times  the  number  sought.  Ans.  11,  or  3^. 

Prob.  22.  It  is  required  to  divide  20  into  two  such  parts, 
that  the  product  of  the  whole  and  one  of  the  parts,  shall  be 
equal  to  the  square  of  the  other. 

Ans.  10v^5  — 10,  and  30—10^5. 

Prob.  23.  A  labourer  dug  two  trenches,  one  of  which  was 
6  yards  longer  than  the  other,  for  111.  I6s.,  and  the  digging 
of  each  of  them  cost  as  many  shillings  per  yard  as  there  were 
yards  in  its  length.     What  was  the  length  of  each  ? 

Ans.  10,  and  16  yards. 

Prob.  24.  A  company  at  a  tavern  had  8/.  15^.  to  pay,  but 
before  the  bill  was  paid,  two  of  them  sneaked  off,  when  those 
who  remained  had  each  lOs.  more  to  pay.  How  many  were 
there  in  the  company  at  first  ?  Ans.  7. 

Prob.  25.  There  are  two  square  buildings,  that  are  paved 
with  stones,  a  foot  square  each.  The  side  of  one  building  ex- 
ceeds that  of  the  other  by  12  feet,  and  both  their  pavements 
taken  together  contain  2120  stones.  What  are  the  lengths 
of  them  separately  ?  Ans.  26,  and  38  feet. 

Prob.  26.  In  a  parcel  which  contains  24  coins  of  silver 
and  copper,  each  silver  coin  is  worth  as  many  pence  as  there 
are  copper  coins,  and  each  copper  coin  is  worth  as  many 
pence  as  there  are  silver  coins,  and  the  whole  is  worth  18 
shillings.     How  many  are  there  of  each  ? 

Ans.  6  of  one,  and  18  of  the  other. 

Prob.  27.  Two  messengers,  A  and  B,  were  despatched  at 
24 


266  SOLUTION  OF  PROBLEMS 

the  same  time  to  a  place  90  miles  distant ;  the  former  of  whom 
riding  one  mile  an  hour  more  than  the  other,  arrived  at  the 
end  of  his  journey  an  hour  before  him.  At  what  rate  did 
each  travel  per  hour  ? 

Ans.  A  went  10,  and  B  9  miles  per  hour. 

Prob.  28.  A  man  travelled  105  miles,  and  then  found  that 
if  he  had  not  travelled  so  fast  by  2  miles  an  hour,  he  should 
have  been  6  hours  longer  in  performing  the  journey.  How 
many  miles  did  he  go  per  hour  ?  Ans.  7  miles. 

Prob.  29.  Bought  two  flocks  of  sheep  for  65/.  135.,  one 
containing  5  more  than  the  other.  Each  sheep  cost  as  many 
shillings  as  there  were  sheep  in  the  flock.  Required  the 
number  in  each  flock.  Ans.  23,  and  28. 

Prob.  30.  A  regiment  of  soldiers,  consisting  of  1066  men, 
is  formed  into  two  squares,  one  of  which  has  4  men  more  in 
a  side  than  the  other.  What  number  of  men  are  in  a  side  of 
each  of  the  squares  ?  Ans.  21,  and  25. 

Prob.  31.  What  number  is  that,  to  which  if  24  be  added, 
and  the  square  root  of  the  sum  extracted,  this  root  shall  be 
less  than  the  original  quantity  by  18  ?  Ans.  25. 

Prob.  32.  A  Poulterer  going  to  market  to  buy  turkeys,  met 
with  four  flocks.  In  the  second  were  6  more  than  three  times 
the  square  root  of  double  the  number  in  the  first.  The  third 
contained  three  times  as  many  as  the  first  and  second  ;  and 
the  fourth  contained  6  more  than  the  square  of  one-third  the 
number  in  the  third  ;  and  the  whole  number  was  1938.  How 
many  were  there  in  each  flock  ? 

Ans.  The  numbers  were  18,  24,  126,  and  1770,  respectively. 

Prob*.  33.  The  sum  of  two  numbers  is  6,  and  the  sum  of 
their  5th  powers  is  1056.     What  are  the  numbers  ? 

^  Ans.  4,  and  2. 

§11.    SOLUTION    OF    problems    PRODUCING    QUADRATIC    EQUA- 
TIONS, INVOLVING  MORE   THAN  ONE   UNKNOWN  QUANTITY. 

366.  It  is  very  proper  to  observe,  that  the  solution  of  a 
problem,  producing  quadratic  equations,  involving  two  un- 
known quantities,  will  sometimes  be  very  much  facilitated  by 
assuming  x  equal  to  their  half  sum,  and  y  equal  to  their  half 
diflference  ;  then,  (Art.  102),  x-\-y  will  denote  the  greater, 
and  x—y  the  lesser.  The  solution,  according  to  this  method 
of  notation,  will,  in  general,  be  more  simple  than  that  which 
would  have  been  found,  if  the  two  unknown  quantities  were 
represented  by  x  and  y  respectively. 


PRODUCING  QUADRATIC  EQUATIONS.    267 

Problem  I.  Required  two' numbers,  such,  that  their  sum, 
their  product,  and  the  difference  of  their  squares,  may  be  all 
equal. 

Let  x-{-i/=  the  greater  ;  and  x—y-=.  the  lesser  ; 

and2a:=:(.x+y)2— (a:— y)2  =  4a:y,  S     ^  ^ 

From  the  2d  equation,  y  =  4  ;  •••  y^— -J  '• 

Now,  by  substituting  this  value  of  y"^,  in  the  first  we  have 
2x=x^-\\  .-.0:2-20:=^,  and  aj^liJV^- 

367.  The  preceding  problem  leads  also  to  the  solution  of 
the  following. 

pROB.  2.  To  find  two  numbers,  such,  that  their  sum,  their 
product,  and  the  sum  of  their  squares,  may  be  all  equal. 

Let,  as  in  the  last  problem,  x-\-y=^  the  greater,  and  x—y=z 
the  lesser  ;  then,  by  the  problem, 
2a;=a;2-y2,  and  2xz^(x-\-yf-\-{x—yf=2x'^-^2y'^  ; 

but  2xz=x'^—y'^  ; 


.-.  by  addition,  3x=2a:2,  and  a;— ^  ; 

.-.  by  substitution,  ^=|  +  3/^  ^  and  y=  :\zW — 3  ; 

.-.  a;;f-y=JJti  V — 3,  and  x — y^iTiV — 3. 
Hence  it  follows,  that  no  two  numbers  can  be  found  to  answer 
the  conditions  ;  and  therefore  the  problem  is  impossible  :  Al- 
though the  above  values  of  x  and  y  are  imaginary,  still  they 
will  satisfy  the  equations,  ^x^zx^—y"^,  and  2x^=z2x^-{-2y'^y 
which  may  be  readily  verified  by  substitution. 

368.  It  is  sometimes  more  expedient  to  represent  one  of 
the  unknown  quantities  by  x,  and  the  other  by  xy-  The 
utility  of  this  method  of  notation  for  eliminating  one  of  the 
unknown  quantities,  will  appear  evident,  from  the  solution  of 
the  following  problem. 

Prob.  3.  What  two  numbers  are  those,  whose  sum  multi- 
plied by  the  greater  is  77  ;  and  whose  difference,  multiplied 
by  the  lesser,  is  equal  to  12  ? 

Let  xyz=.  the  greater,  and  x^=i  the  lesser  ;  then  by  the  pro- 
blem, x'^y''--{-xy~ll,  and  x'^y—x'^  —  \2  ; 

2         77  ,    ,        12  12  77 

••  a;^=~- — ,  and  x^z=z- 


y-'-\-y  y-1'  "y-i      y^^y' 

and  clearing  of  fractions,  12y2-|-i2y=77y— 77  ; 

by  transposition  and  division,  y^ _y_ ; 


268  SOLUTION  OF  PROBLEMS 

.'.completing  the  square,  and  extracting  the  root,  y=y,  or 

J.     Either  value  of  y  will  answer  the  conditions  of  the  prob- 

12 
lem  ;  Lety=zJ;   then  a?=: --=16;    .•.a?=:^4,  and  a:y=: 

±7.  Hence  the  numbers,  by  taking  the  positive  values,  are 
4  and  7.  Let  also  y  =  V  ;  then  x^—\\  .-.0?=: -tf  .^2,  and 
a:yr=y  X  ifV^  — i  V  V  2-  Hence  the  irrational  numbers, 
\^'l  and  y  -^2,  will  also  answer  the  conditions  of  the  prob- 
lem. 

369.  When  a  problem  expresses  more  than  two  distinct 
conditions,  which  require  to  b^  translated  into  as  many  equa- 
tions ;  the  solution  cannot  be  obtained  by  means  of  quadra- 
tics, unless  that  some  of  the  equations  are  of  the  first  degree  ; 
for  the  final  equation  resulting  from  the  elimination  of  the 
imknown  quantities  will,  in  general,  be  of  a  higher  degree  than 
the  second.  There  are,  however,  some  particular  cases  in 
which  the  unknown  quantities  may  be  eliminated  by  certain 
artifices,  (which  are  best  learned  by  experience),  so  as  to  leave 
the  final  equation  of  a  quadratic  form. 

Prob.  4.  It  is  required  to  find  three  numbers,  such,  that 
the  product  of  the  first  and  second,  added  to  the  sum  of  their 
squares,  shall  be  equal  to  37  ;  the  product  of  the  first  and  third 
added  to  the  sum  of  their  squares,  shall  be  equal  to  49  ;  and 
the  product  of  the  second  and  third  added  to  the  sum  of  their 
squares,  shall  be  equal  to  61. 

Let  a;=  the  first  number,  3/=  the  second,  and  z=  the  third 
Then,  x^+y'^-{-xy='^l ;  \ 

x'^-\-2:^-\-xz=49  ;  >  by  the  problem. 

and  y2_|_2:2_[_y;3  — 61  ;  ) 

By  subtracting  the  first  equation  from  the  second,  x^—y^-\- 

12 
(z-'y)x  =  l2;.'.z-^y-\-x=^— (a). 

By  subtracting  the  second  equation  from  the  third,  y^—x^+ 

12 
(y^x)z=l2;.'.y+x-[-z=-— (b); 

•••jz:^=^'^"^y-^=^-y;  .•.2y=^+^. 

By  substitutrng  2y  for  x-\-z,  in  equations  (a)  and  (i),  we 

^   j«         12  ^  „         12 

find  3y= ,  and  3y= 


y  y-x 

zy — y'^=:4,  and  y^—yx=4  ; 

y2+4       -       y^— 4         n    ly^ 
z=/- ,  and  a;=^^ ;  /. -^—z:' 


y 


r^): 


PRODUCING  QUADRATIC  EQUATIONS.    269 

Now,  by  substituting  these  values  of  x  and  x"^  in  the  first  of 
the  original  equations,  it  becomes 

(7/2  —  4v2                     v^  —  4 
)  ■\-y'^-\-y'- =37;  .-.by reduction, 

49 
y* — — y2=_16  ;  and,  by  completing  the  square, 
o 

^     49  ,  ,   /49\2     2401-192  ,     49,47 

y  -yy  +(t)  =—36 — =  •■•*  =-6-^-6-' 

and,  by  taking  the  positive  sign,  y=-]-4  ; 

y2_4        16—4 

.'.by  taking  y=:4,  x=.- — = =3,  and 

_yH-4__16-H_20__ 

~   y    ~~    4    ~~  4  ~  * 

Hence  the  three  numbers  sought  are  3,  4,  and  5,  which  are 
in  arithmetical  progression.  This  relation  appears  also  evi- 
dent from  the  result  2y^=.x-\-z^  found  in  the  beginning  of  the 
solution. 

Prob.  5.  There  are  three  numbers,  the  difference  of  whose 
differences  is  8  ;  their  sum  is  41  ;  and  the  sum  of  their  squares 
669.     What  are  the  numbers  ? 

Let  a;=:  the  second  number, 
and  y=  the  difference  of  the  second  and  least ; 
.-.  x~y,  X,  and  x+y  +  S  are  the  numbers, 
and  their  sum  =:3a:-f  8  =  41  ;  .*.  3a:=33,  and  a;=rll  ; 
.•.(ll-y)2  +  r214-(19  +  y)2  =  669,  or  y2_|_ 8^=48  ; 
.-.  completing  the  square*  and  extracting  the  root, 
y  +  4=  ±8,  .and  y  =  4,  or — 12,  both  which  values  answer 
the  conditions  ;  and  the  numbers  are  7,  11,  and  23. 

Prob.  6.  What  number  is  that,  which  being  divided  by  the 
product  of  its  two  digits,  the  quotient  is  2  ;  and  if  27  be  added 
to  it,  the  digits  will  be  inverted  ?  Ans.  36. 

Prob.  7.  There  are  three  numbers,  the  difference  of  whose 
differences  is  5  ;  their  sum  is  44  ;  and  continual  product  is 
1950.     What  are  the  numbers  ?  Ans.  6,  13,  and  25. 

Prob.  8.  A  farmer  received  71.  4s.  for  a  certain  quantity 
of  wheat,  and  an  equal  sum  at  a  price  less  by  1^.  6c?.  per 
bushel,  for  a  quantity  of  barley,  which  exceeded  the  quantity 
of  wheat  by  16  bushels.  How  many  bushels  were  there  of 
each  ?  Ans.  32  bushels  of  wheat,  and  48  of  barley. 

Prob.  9.  A  poulterer  bought  15  ducks  and  12  turkeys  for 
five  guineas.     He  had  two  ducks  more  for  18  shillings,  than 
24* 


270  SOLUTION  OF  PROBLEMS 

he  had  of  turkeys  for  20  shillings.     What  was  the  price  of 
each  ?     Ans.  the  price  of  a  duck  was  3^.  and  of  a  turkey  5^. 

Prob.  10.  There  are  three  numbers,  the  difference  of  whose 
differences  is  3  ;  their  sum  is  21  ;  and  the  sum  of  the  squares 
of  the  greatest  and  least  is  137.     Required  the  numbers. 

Ans.  4,  6,  and  11. 

Prob.  ]  1.  There  is  a  number  consisting  of  2  digits,  which, 
when  divided  by  the  sum  of  its  digits,  gives  a  quotient  greater 
by  2  than  the  first  digit.  But  if  the  digits  be  inverted,  and 
then  divided  by  a  number  greater  by  unity  than  the  sum  of  the 
digits,  the  quotient  is  greater  by  2  than  the  preceding  quotient. 
Required  the  number.  Ans.  24. 

Prob.  12.  What  two  numbers  are  those,  whose  product  is 
24,  and  whose  sum  added  to  the  sum  of  their  squares  is  62  ? 

Ans.  4,  and  6. 

Prob.  13.  A  grocer  sold  80  pounds  of  mace,  and  100 
pounds  of  cloves,  for  65/.  ;  but  he  sold  60  pounds  more  of 
cloves  for  201.  than  he  did  of  mace  for  10/.  What  was  the 
price  of  a  pound  of  each  1 

Ans.  the  mace  cost  10^.  and  the  cloves  5^.  per  pound. 

Prob.  14:  To  divide  the  number  134  into  three  such  parts, 
that  once  the  first,  twice  the  second,  and  three  times  the  third, 
added  together,  may  be  equal  to  278  ;  and  that  the  sum  of  the 
squares  of  the  three  parts  may  be  equal  to  6036. 

Ans.  40,  44,  and  50,  respectively. 

Prob.  15.  Find  two  numbers,  such,  that  the  square  of  the 
greater  minus  the  square  of  the  lesger,  may  be  56  ;  and  the 
square  of  the  lesser  plus  one  third,  their  product  may  be  40. 

Ans.  9,  and  5. 

Prob.  16.  There  are  two  numbers,  such,  that  three  times 
the  square  of  the  greater  plus  twice  the  square  of  the  less  is 
110 ;  and  half  their  product,  plus  the  square  of  the  lesser,  is 
4.     What  are  the  numbers?  Ans.  6,  and  1. 

Prob.  17.  What  number  is  that,  the  sum  of  whose  digits  is 
15  ;  and  if  31  be  added  to  their  product,  (he  digits  will  be  in- 
verted 1  Ans.  78. 

Prob.  18.  There  are  two  numbers,  such,  that  if  the  lesser 
be  taken  from  three  times  the  greater,  the  remainder  will  be 
35  ;  and  if  four  times  the  greater  be  divided  by  three  times  the 
lesser  plus  one,  the  quotient  will  be  equal  to  the  lesser  number. 
What  arc  the  numbers  ?  Ans.  13,  and  4. 

Prob.  19.  To  find  two  numbers,  the  first  of  which, /)/u5 
2,  multiplied  into  the  second,  minus  3,  may  produce  110  ;  and 


PRODUCING  QUADRATIC  EQUATIONS.    271 

the  first  minus  3,  multiplied  by  the  second  plus  2,  may  pro- 
duce 80.  Ans.  8,  and  14. 
Prob.  20.  Two  persons,  A  and  B,  comparing  their  wages, 
observe,  that  if  A  had  received  per  day,  in  addition  to  what 
he  does  receive,  a  sum  equal  to  one-fourth  of  what  B  receiv- 
ed per  week,  and  had  worked  as  many  days  as  B  received 
shillings  per  day,  he  would  have  received  48.y.  ;  and  had  B 
received  2  shillings  a  day  more  than  A  did,  and  worked  for  a 
number  of  days  equal  to  half  the  number  of  shillings  he  re- 
ceived per  week,  he  would  have  received-4Z.  \8s.  What  were 
their  daily  wages  1                   Ans.  A's  5  shillings,  and  B's  4. 

Prob.  21.  Bacchus  caught  Silenus  asleep  by  the  side  of  a 
full  cask,  and  seized  the  opportunity  of  drinking,  which  he 
continued  for  two-thirds  of  the  time  that  Silenus  would  have 
taken  to  empty  the  whole  cask.  After  that  Silenus  awoke, 
and  drank  what  Bacchus  had  left.  Had  they  drunk  both 
together,  it  would  have  been  emptied  two  hours  sooner,  and 
Bacchus  would  have  drunk  only  half  what  he  left  Silenus.  Re- 
quired the  time  in  which  they  could  have  emptied  the  cask 
separately.  Ans.  Silenus  in  3  hours,  and  Bacchus  in  6. 

Prob.  22.  Two  persons,  A  and  B,  talking  of  their  money, 
A  says  to  B,  if  I  had  as  many  dollars  at  ^s.  6rf.  each,  as  I 
have  shillings,  I  should  have  as  much  money  as  you  ;  but,  if 
the  number  of  my  shillings  were  squared,  I  should  have  twice 
as  much  as  you,  and  12  shillings  more.     What  had  each  ? 

Ans.  A  had  12,  and  B  66  shillings. 

Prob.  23.'^t  is  required  to  find  two  numbers,  such,  that  if 
their  product  be  added  to  their  sum  it  shall  make  62  ;  and  if 
their  sum  be  taken  from  the  sum  of  their  squares,  it  shall  leave 
86.  Ans.  8,  and  6. 

Prob  24.  It  is  required  to  find  two  numbers,  such,  that 
their  difference  shall  be  98,  and  the  diff'erence  of  their  cube 
roots  2.  Ans.  125,  and  27. 

Prob.  25.  There  is  a  number  consisting  of  two  digits.  The 
left-hand  digit  is  equal  to  3  times  the  right-hand  digit  ;  and  if 
12- be  subtracted  from  the  number  itself,  the  remainder  will 
be  equal  to  the  square  of  the  left-hand  digit.  What  is  the 
number  ?  Ans.  93. 

Prob.  26.  A  person  bought  a  quantity  of  cloth  of  two  sorts 
for  11.  18  shillings.  For  every  yard  of  the  be,tter  sort  he  gave 
as  many  shillings  as  he  had  yards  in  all  ;  and  for  every  yard 
of  the  worse  as  many  shillings  as  there  were  yards  of  the  bet- 


272        EXPANSION  OF  INFINITE  SERIES. 

ter  sort  more  than  of  the  worse.  And  the  whole  price  of 
the  better  sort  was  to  the  whole  price  of  the  worse  as  72  to  7, 
How  many  yards  hadr  he  of  each  ? 

Ans.  9  yards  of  the  better,  and  7  of  the  worse. 
Prob.  27.  There  are  four  towns  in  the  order  of  the  let- 
ters, A,  B,  C,  D.  The  difference  between  t!ie  distances,  from 
A  to  B,  and  from  B  to  C,  is  greater  by  four  miles  than  the  dis- 
tance from  B  to  D.  Also  the  number  of  miles  between  B 
and  D  is  equal  to  ,two-thircl3  of  the  number  between  A  to  C. 
And  the  number  between  A  and  B  is  to  the  number  between 
C  and  D  as  seven  times  the  number  between  A  and  C  :  26. 
Required  the  respective  distances. 

Ans.  AB=42,  BC=6,  and  CD=26  miles. 


CHAPTER  XII. 

ON 

THE   EXPANSION   OF   INFINITE    SERIES. 

§  I.  RESOLUTIONS  OF  ALGEBRAIC  FRACTIONS. 

370.  An  infinite  series  is  a  continued  rank,  or^ogression  of 
quantities,  connected  together  by  the  signs  +  or  —  ;  and  usu- 
ally proceeds  according  to  some  regular,  or  determined  law. 

Thus,  i  +  i+J+i+l+^V+.  ^^' 

In  the  first  of  which,  the  several  terms  arc  the  reciprocals 
of  the  odd  numbers,  1,  3,  5,  7,  &c. ;  and  in  the  latter  the  recipro- 
cals of  the  even  numbers,  2,  4,  6,  8,  &c.,  with  alternate  signs. 

371.  We  have  already  observed  (Art.  96),  that  if  the  first 
or  leading  term  of  the  remainder,  in  the  division  of  algebraic 
quantities,  be  not  divisible  by  the  divisor,  the  operation  might 
be  considered  as  terminated  ;  or,  which  is  the  same,  that  the 
integral  part  of  the  quotient  has  been  obtained.  And  it  has 
also  been  remarked,  (Art.  89),  that  the  division  of  the  remain- 
der by  the  divisor  can  be  only  indicated,  or  expressed,  by  a 
fraction:  thus,  for  example,  if  we  have  to  divide  a'  by  a-f-1, 


i 


EXPANSION  OF  INFINITE  SERIES.        273 

we  write  for  the  quotient  — -—  :  This,  however,  does  not  pre- 

a-f- 1 

vent  us  from  attempting  the  division  according  to  the  rules  that 
have  been  given,  nor  from  continuing  it  as  far  as  we  please,  and 
we  shall  thus  not  fail  to  find  the  true  quotient,  though  under 
different  forms. 

372.  To  prove  this,  let  us  actually  divide  a"  or  1,  by  1— a, 
thus ; 

•       I -a 


I 
I— a 

remainder     a 


Quot.  1+-   ^ 


1  +  a 

Therefore- =1+:; ;but- =a-\-- ;    ■- = 

1 — a  I— a  I— a  I — a       1 — a 

1 — a      1 — a  1  —  a     1 — a  1 — a 

This  shows  that  the  fraction may  be  exhibited  under 

1 — a       "^ 

all  the  following  forms  : 

la  d^ 

1— a  1 — a  I — a 


1-a'  '      ^      ^1— a' 

1  — a 
Now,  by  considering  the  first  of  these  formulae,  which  is 

l+i ,  and  observing  that  1  =:- ,   we   have   1+- =: 

1 — a  °  1 — a  1 — a 

1— a        a     _1— a-f-a_     1 
1 — a     l—a~    1— a         1 — a 

If  we  follow  the  %ame  process  with  regard  to  the  second  ex- 
pression, that  is  to  say,  if  we  reduce  the  integral  part  l  +  a  to 

the  same  denominator,  1  —a,  we  shall  have  the  fraction , 

1  — a 

to  which  if  we  add ,  we  shall  have =- • 

1 — a  \—a         i—a 

.     In  the  third  formula  of  the  quotient,  the  integers  l+a+«^ 

\ flS 

reduced  to  the  denominator  1  —a  make  ,  and  if  we  add 

1  — a 

a^  1 

to  it  the  fraction the  sum  will  be 


I— a  I 


274        EXPANSION  OF  INFINITE  SERIES. 

Therefore  each  of  these  formula  is  in  fact  the  value  of  the 

proposed  fraction . 

■^     ^  1 — a 

273.  This  being  the  case,  we  may  continue  the  series  as  far 
as  we  please,  without  being  under  the  necessity  of  performing 
any  more  calculations  ;  by  observing,  in  the  first  place,  that 
each  of  these  formulae  is  composed  of  an  integral  part  which 
is  the  sum  of  the  successive  powers  of  a,*beginning  with  a^=z\ 
inclusively ;  * 

Secondly,  of  a  fraction  which  has  always  for  the  denomi- 
nator 1 — a,  and  for  the  numerator  the  letter  c,  with  an  ex- 
ponent greater,  by  unity,  than  that  of  the  same  letter  in  the 
last  term  of  the  integral  part. 

This  constant  formation  of  the  successive  formulae,  is  what 
Analysts  call  a  law.  And  the  manner  of  deducing  general 
laws  by  the  consideration  of  certain  particular  cases,  is  usu- 
ally called  induction  ;  which,  though  not  a  strict  method  of 
proof,  says  Laplace,  has  been  the  source  of  almost  all  the 
discoveries  that  have  hitherto  been  made,  both  in  analysis  and 
physics,  of  which  all  the  phenomena  are  the  mathematical  re- 
sults of  a  small  number  of  invariable  laws.  It  is  thus  that 
Newton,  by  following  the  law  of  the  numeral  coefficients,  in 
the  square,  the  cube,  the  fourth  power,  &c.  of  a  binomial, 
arrived  soon  at  the  general  law,  that  is  to  say,  at  the  general 
formula  that  bears  his  name,  and  which  will  be  demonstrated 
in  one  of  the  following  Sections  :  This  Geometer  has  carefully 
added,  that  in  following  this  mode  of  investigation,  we  must 
not  generalize  too  hastily  ;  as  it  often  happens,  that  a  law, 
which  appears  to  take  place  in  the  first  part  of  a  process,  i^ 
not  found  to  hold  good  throughout.      Thus,  in  the  simple  in- 

,      ,     .        531251  ^     .      ,    . 

stance  of  reducmg  to  a  decimal,  its  equivalent  value 

is  17174949,  &c.,  of  which  the  real,  repeating  period  is  49, 
and  not  17,  as  might,  at  first,  be  imagined. 

374.  From  what  has  been  observed  with  regard  to  the  suc- 
cessive quotients,  we  can,  in  general,  put 

--i-  =  H-a  +  «2:|-a3+a*       . """-^TI^^ 

I— a  l-ha 

n  being  a  whole  positive  number,  which  augmented  by  unity, 

gives  the  place  of  the  term.     In  fact,  making  w=:3,  a"  becomes* 

a^,  which  is  the  fourth  term  of  the  quotient,  for  n  =  4,  cr"  becomes 

c*,  which  is  the  fifth  term.     But  as  nothing  hinders  us  from 

removing  indefinitely  the  fractional  term  which  terminates  the 

series,  that  is,  of  adding  always  a  term  to  the  integral  part ; 


EXPANSION  OF  INFINITE  SERIES.        275 

so  that  we  might  stiil  go  on  without  end  ;  for  which  reason  it 
may  be  said  that  the  proposed  fraction  has  been  resolved  into 
an  infinite  series  ;  which  is,  l-}-a-\-a'^-^a'^-{-'^-^a^-\-a^-{-a''-\-a^ 
-\-a^-{-a^^-\-a^^-\-a^'^-\-,  &c.4o  infinity:  and  there  are  sufli- 
cient  grounds  to  maintain  that  the  value  of  this  infinite  series 

is  the  same  as  that  of  the  fraction 

I— a 

Orthat,-i-=14-a+a2-j-a3_j_a4_|- .   &c. 
1— a 

375.  What  has  been  just  observed  may  at  first  appear 
strange  ;  but  the  consideration  of  some  particular  cases  will 
make  it  easily  understood. 

Let  us  suppose,  in  the  first  place,  a  =  l  ;  the  general  quo- 
tient above  will  become  a  particular  quotient  corresponding 

to  the  fraction  - — -.     The  series  taken  indefinitely,  shall  be 

^=14-1  +  1  +  1  +  1  +  1+,  &c. 

In  order  to  seie  clearly  the  meaning  of  this  result,  let  us 
suppose  that  we  have  to  divide  unity  or  1  successively  by  the 

numbers  1.  ^,   ^-,    J^,    ^,  &c.,  we  will  have  the 

quotient,  1,  10,  100,  1000,  10000,  &c.,  continually  and  inde- 
finitely increasing  ;  because  the  divisors  are  continually  and 
indefinitely  decreasing  ;  but  these  divisors  tend  towards  zero, 
which  they  cannot  attain,  although  they  approach  to  it  con- 
tinually, or  that  the  difference  becomes  less  and  less  ;  and  at 
the  same  time  the  value  of  the  fraction  increases  continually, 
and  tends  to  that  which  correspo.nds  to  the  divisor  zero  or  0  ; 
and  it  is  as  much  impossible  that  the  fraction  in  its  successive 

augmentations,  attains  -,  as  it  is  that  the  denominator  in  its 

successive  diminutions  arrives  at  zero.  Thus  -  is  the  last 
term  or  limit  of  the  increasing  values  of  the  fraction  :  at  this 
period,  it  has  received  all  its  augmentations  :  -  is  not  therefore 
a  number,  it  is  the  superior  limit  of  numbers  ;  such  is  the  no- 
tion that  we  must  have  of  this  result  -,  which  the  analysts  call, 

for  abbreviation,  infinitt/,  and  which  is  denoted  by  the  character 
00,  (Art.  35).     It  is  frequently  given  as  an  answer  to  an  im- 


276        EXPANSION  OF  INFINITE  SERIES. 

possible  question,  (which  will  be  noticed  in  a  subsequent  part 
of  the  Work) ;  and  in  fact,  it  is  very  proper  to  announce  this 
circumstance,  since  that  we  cannot  assign  the  number  denoted 
by  this  sign. 

It  may  still  be  remarked,  that  if  we  would  take  but  the  first 
six  terms  of  the  series,  we  must  close  the  development  by  the 
corresponding  remainder  divided  by  this  divisor,  which  gives, 

:^4=l  +  l  +  l  +  l  +  l  +  l+i; 

this  equality,  absurd  in  appearance,  proves  that  six  terms  at 
least  do  not  hinder  the  series  from  being  indefinitely  conti- 
nued. And  in  fact,  if  after  having  taken  away  six  terms  from 
this  series,  it  would  cease  to  be  infinite,  or  become  terminat- 
ed, in  restoring  to  it  these  six  terms,  it  should  be  composed  of 
a  definite  or  assignable  number  of  terms,  which  it  is  not. 
Therefore  the  surplus  of  the  series  must  have  the  same  sum 

as  the  total.     We  can  yet, say  that  -,  inasmuch  as  it  is  not 

a  magnitude,  can  receive  no  augmentation,  so  that  1  -\-.l  -}- 1  + , 

Slc.  +-  must  remain  equal  to  -. 

Hence,  we  might  conclude  that  a  finite  quantity  added  to, 
or  subtracted  from  infinity,  makes  no  akeration. 

Thus,    QO-l-azz:  c». 

However,  it  may  be'necessary  in  this  place  to  observe,  that, 
although  an  infinity  cannot  be  increased,  or  decreased,  by  the 
addition,  or  subtraction,  of  finite  quantities  ;  still,  it  may  be  in- 
creased or  decreased,  by  multiplication  or  division,  in  the  same 

manner  as  any  other  quantity ;  Thus,  if-  be  equal  to  infinity, 

2  3 

-  will  be  the  double  of  it,  -  thrice,  and  so  on.     See  Euler's 

Algebra,  Vol.  I. 

Note. —  -,  -j-,  — j— ,  — ^ — ,  &c.  are  considered  to  be  frac- 
1    10   Too"   lOoo 

tions,  in  which  the  denominators  are  1,  -y-,  -^,  — j — ,  &c. 

To    TWU    ToTTTT 

Now,  as  1  divided  by  any  assignable  quantity,  however 
great  it  may  be,  can  never  arrive  completely  at  0,  consequent- 
ly the  fractions  in  their  successive  augmentations  can  never 
arrive  at  infinity,  except  that  unity  or  1,  be  divided  by  a 
quantity  infinitely  great ;  that  is  to  say,  it  must  be  divided  by 


EXPANSION  OF  INFINITE  SERIES.        27T 

infinity ;  hence  we  may  conclude  that  -^  is  in  reality  equal  to 

nothing,  or  oo-=0. 

360.  It  may  not  be  improper  to  take  notice  in  this  place  of 
other  properties  of  nought  and  infinity. 

I.  That  nought  added  to  or  subtracted  from  any  quantity, 
makes  it  neither  greater  nor  less  ;  that  is, 

a-J-0=a,  and  a — 0=a. 

II.  Also,  if  nought  be  multiplied  or  dirided  by  any  quantity, 
both  the  product  and  quotient  will  be  nought ;  because  any 
number  of  times  0,  or  any  part  of  0,  is  0  :  that  is, 

Ox«,  or  ax 0  =  0,  and  -=0. 
a 

III.  From  the  last  property,  it  likewise  follows,  that  nought 
divided  by  nought,  is  a  finite  quantity,  of  some  kind  or  other. 
For  since  0  X  a=0,  or  0=^0  x  a,  it  is  evident  from  the  ordinary 
rules  of  division,  that 

0 
0  =  ^- 

IV.  Farther,  if  nought  be   multiplied  by  infinity,  the  pro- 

.1        a 
duct  will  be  some  infinite  quantity.     For  since  -  or  -=oo  ; 

therefore,  0  x  oo  =a. 

361.  It  may  be  also  remarked,  that  nought  multiplied  by  0 
produces  0  ;  that  is, 

OxOr^O. 
For,  since  0  x  «  =  0,  whatever  quantity  a  may  be,  then,  sup- 
posing a  =  0,  0X0  =  0. 

From  this  we  might  infer,  according  to  the  rules  of  division, 

that  the  value  of --=:0,  or  that  nought  divided  by  nought  is 

nought,  in  this  particular  case. 

Also,  that  0,  raised  to  any  power,  is  0  ;  that  is,  0'«=0  ;  it 

Q^      0  w^ 

follows  that  -—  =- :  but  if  in  <2"»-'»=—  (Art.  86),  we  suppose 

c=:0,  which  may  be  allowed,  since  a  designates  any  number, 

we  have  0®=-. 

If  we  really  effect  the  division  of  0  by  0,  we  could  put  for 
the  quotient  any  number  whatever,  since  any  number,  multi- 
plied by  zero,  gives  for  the  product  zero,  which  is  here  the 
dividend. 

25 


278-       EXPANSION  OF  INFINITE  SERIES. 

This  expression,  O**,  appears  therefore  to  admit  of  an  infi- 
nity of  numerical  values  ;  and  yet  such  a  result  as  -  can,  in 
many  cases,  admit  of  a  finite  and  determined  value.  It  is  thus, 
foi  example,  that  the  fraction ,  in  the  hypothesis  of  a=0, 

,  Kxo   6 

becomes  — - — =-. 

But,  if  at  first  we  write  this  fraction  under  the  form  Ka"*-"*, 
and  that  we  put  a=0,  we  find  that  it  becomes  KxO*"~", 
which  is  0  for  wi>7» ;  in  case  of  m<^n,  or  m=n  —  d,  we  shall 

rr        rr 

have  (Art.  86),  -frj=-^ ;  which  is  equal  to  infinity,  as  has  been 

already  observed ;  finally,  for  m=n,  we  can  divide  above  and 
below  by  a*",  and  the  fraction  is  reduced  to  K,  which  is  a  finite 
quantity. 

362.  If  we  suppose,  in  the  fraction  (Art.  358),  a=2,  we 
find 

,-^  =  1+2  +  4  +  8+16  +  32  +  64  +  ,  &c., 

which  at  first  sight  it  will  appear  absurd.  But  it  must  be  re- 
marked, that  if  we  wish  to  stop  at  any  term  of  the  above  se- 
ries, we  cannot  do  so  without  joining  the  fraction  which  re- 
mains. Suppose,  for  example,  we  were  to  stop  at  64  ;  after 
havmg  written  1+2+4+8+16+32  +  64;  we  must  join  the 

128         128 
fraction  - — -,  or — -,  or  —128  ;  we  shall  therefore  have  for 
1  —2        —  1 

the  complete  quotient  127  —  128,  than  is  in  fact  —1. 

Here,  however  far  the  fractional  term  may  be  extended,  its 
numerical  value,  which  is  negative,  will  always  surpass,  by  a 
unit,  that  of  the  integral  part,  so  that  this  is  totally  destroyed  ; 
and  as  in  the  hypotheses  of  a>l,  we  shall  always  subtract 
more  than  what  we  will  add,  we  shall  never  meet  with  the 

result  -. 

363.  These  are  the  considerations  which  are  necessary 
when  we  assume  for  a  numbers  greater  than  unity  ;  but  if 
we  now  suppose  a  less  than  1,  the  whole  becomes  more  in- 
telligible ;  for  example,  let  a=^,  and  we  shall  have = 

- — r-=Y=2,  which  will  also  be  equal  to  the  following  se- 
1— ^      7 


EXPANSION  OF  INFINITE  SERIES.        279 

ries,  l+i+i+i:4-TV+3^2+^+Tl8.  &c.,  to  infinity  (Art. 
358).  Now,  if  we  take,  only  two  terms  of  the  series,  we 
shall  have  1+i,  and  it  wants -J  .of  being  equal  to  2  ;  if  we 
take  three  terms,  it  wants  1,  for  the  sum  is  1|- ;  if  we  take 
four  terms,  we  have  i-J,  and  the  deficiency  is  only  J  :  There- 
fore, we  see  very  clearly  that  the  more  terms  of  the  quotient 
we  take,  the  less  the  difference  becomes ;  and  that,  conse- 
quently, if  we  continue  to  take  successive  portions  of  this 
series,  the  differences  between  those  consecutive  sums  and 

the  fraction f =2,  decrease,  and  end  by  becoming  less  than 

any  given  number,  however  small  it  may  be.  The  number 
2  is  therefore  still  a  limit,  according  to  the  acceptation  of  this 
word. 

Now,  it  may  be  observed,  that  if  the  pieceding  series  be 
continued  to  infinity,  there  will  be  no  difference  at  all  between 

its  sum  and  the  value  of  the  fraction j-,  or  2, 

364.  A  limit,  according  to  the  notion  of  the  ancients,  is  some 
fixed  quantity,  to  whioh  another  of  variable  magnitude  can  never 

become  equal,  though,  in  the  course  of  its  variation,  it  may  ap- 
proach nearer  to  it  than  any  difference  that  can  be  assigned  ; 
always  supposing  that  the  change,  which  the  variable  quantity 
undergoes,  is  one  of  contimced  increase,  or  continued  diminution 
Such,  for  example,  is  the  area  of  a  circle,  with  regard  to  the 
areas  of  the  circumscribed  and  inscribed  polygons,  for,  by  in- 
creasing the  number  of  sides  of  these  figures,  their  difference 
may  be  made  less  than  any  assigned  area,  however  small ; 
and  since  the  circle  is  necessarily  less  than  the  first,  and 
greater  than  the  second,  it  must  differ  from  either  of  them  by 
a  quantity  less  than  that  by  which  they  differ  from  each  other. 
The  circle  will  thus  answer  all  the  conditions  of  a  limit, 
which  is  included  in  the  above  definition. 

365.  The  preceding  considerations  are  very  proper  to  de- 
fine the  nature  of  the  word  limit ;  but  as  algebra,  which  is 
the  subject  we  are  treating  of  here,  needs  no  foreign  aid  to 
demonstrate  its  principles,  it  is  necessary,  therefore,  to  explain 
the  nature  of  the  word  limit,  by  the  consideration  of  algebraic 
expressions.      For  this  purpose,  let,  in  the  first  place,  the 

very  simple  fraction  be ,  in  which  we  suppose  that  x  may 

x-\-a 
be  positive,  and  augmented  indefinitely  ;  in  dividing  both  terms 

of  this  fraction  by  x,  the  result  -r-j;T^   evidently    shows    that 


280        EXPANSION  OF  INFINITE  SERIES. 

the  function  remains  always  less  than  a,  but  that  it  approaches 
continually  to  a,  since  that  the  part  ^,  of  its  denominator,  di 

minishes  more  and  more,  and  can  be  reduced  to  such  a  degree 
of  smallness  as  we  would  wish. 

366.  The  difference  between  a  and  the  proposed  fraction  be- 
ing in  general  expressed  by  a — -= — ; — ,  becomes  so  much 

x-\-a     x-\-a 

smaller,  according  as  x  is  larger,  and  can  be  rendered  less 
than  any  given  magnitude,  however  small  it  may  bo  ;  so  that 
the  proposed  fraction  can  approach  to  a  as  near  as  we  would 

QX 

wish  :  a  is  therefore  the  limit  of  the  fraction ,  relatively  to 

,  x-\-a  '' 

the  indefinite  augmentation  which  x  can  receive.  It  is  in  the 
characters  which  ^e  have  just  expressed,  that  the  true  accep- 
tation, which  we  must  give  to  the  v/ord  limit,  consists,  in  order 
te  comprehend  every  thing  which  can  relate  to  it. 
.  367.  If  we  had  remarked  in  the  preceding  example,  that  by 
carrying  on,  as  fa#as  we  would  wish,  the  augmentation  of  a?,  we 

could  never  regard,  as  nothing,  the  fraction  — ; —  ;    therefore 

x-\-a 

QX 

we  would  reasonably  conclude,  that  the  fraction  — - — ,  though 

x-\-a 

it  would  approach  indefinitely  to  the  limit  a,  could  never  at- 
tain a,  and,  consequently,  cannot  surpass  it  ;  but  it  would  be 
wrong  to  insert  this  circumstance  as  a  condition  in  the  gene- 
ral definition  of  the  word  limit ;  we  would  thereby  exchide 
the  ratios  of  vanishing  quantities,  ratios  whose  existence  is 
incontestable,  and  from  which  we  derive  much  in  analysis. 

368.  In  fact,  when  we  compare  the  functions  ax  and  ax-\- 
a?^,  we  find  th&t  their  ratio,  reduced  to  its  most  simple  ex- 

pfession,  is  — ■ — ,  and  that  it  approaches  nearer  and  nearer  to 
a-\-x  ^'^ 

unity,  according  as  x  diminishes.  It  becomes  exactly  1 ,  when 
x=.Q  ;  but  the  quantities  ax  and  ax-^-x"^^  which  are  then  rigor- 
ously nothing,  can  they  have  a  determinate  ratio  ?  This  is 
what  appears  diflficult  to  conceive  ;  and  we  cannot  give  a  clear 
idea  of  it  but  by  presenting  the  quantity  1  as  a  limit  to  which 
.the  ratio  of  the  functions  ax  and  ax-^-x"^  can  approach  as  near 

a  x 

as  we  would  wish,  since  the  difference,  1 ; — = — ; — ,  can 

a  +  x     a-{-x 

be  rendered  less  than  any  assignable  magnitude,  however 
small  this  magnitude  may  be. 


EXPANSION  OF  INFINITE  SERIES.  281 


On  the  other  hand,  the  ratio  — ; — ,.of  the  quantities  ax  and 

a-\-x 

ax-^-x"^  can  not  only  attain  unity  when  we  make  x=0,  but 
surpass  it  when  we  suppose  x  negative,  since  it -becomes  then 

,  a  quantity  which  is  greater  than  1,  when  x<Ca.     This 

circumstance  appears  not  at  all  contrary  to  the  idea  of  limit ; 
for  we  can  regard  the  value  1,  which  answers  to  a;=0,  as  a 
term  towards  which  the  ratio  of  the  functions  ax  and  ax-{-x'^' 
tends,  by  the  diminutions  of  the  values  of  x,  whether  positive 
or  negative.  For  further  illustrations  of  the  world  limit,  and 
what  is  meant  by  infinity,  and  infinitely  small  quantities  or  in- 
finitesimals, the  intelligent  reader  is  referred  to  Lacroix's 
Introduction  to  the  Traite  du  Calcul  Differentiel  et  du  Calcul  In- 
tegral,  4to.  where  these  subjects  are  clearly  elucidated. 

369.  Now,  let  a=^,  in  the  fraction ,  and    we    shall 

have  ^_L_^3^i_|.i_^^_^^i__{._i^^_l__^^  &c.     If  we  take 

^         3 

two  terms,  we  find  1-f-J,  and  the  difference  =i ;  three  terms 
give  1+J,  the  error  =^;  for  four  terms  the  error  is  no  more 
than  -}f.  Since,  therefore,  the  error  always  becomes  three 
times  less,  it  tends  towards  zero,  which  it  cannot  attain,  and 
the  sum  tends  toward  |-,  which  is  the  limit. 

370.  Again,  let  us  take  a=^^,  and  we  shall  have- — ^  =3 

=  l+f+l  +  2y+if+3^A+&c.;  here,  in  the  first  place, 
the  sum  of  two  terms,  which  is  l  +  f»  is  less  than  3  by  1+^- ; 
taking  three  terms,  which  make  2j,  the  error  is  f  ;  for  four 
terms,  wiiose  sum  is  2ji  the  error  is  ^. 

371.Finally,fora=l,wefind--i-5-=l+i=H-i+^+A 

+  25-6  +  '  ^^'  ?  ^^^  fi^st  two  terms  are  equal  to  Ij,  which  gives 
■j?2  for  the  error  ;  and  taking  one  term  more,  we  shall  have 
only  an  error  of  -Jg-. 

372.  From  the  preceding  considerations  we  may  readily 
conclude,  that  any  fraction  having  a  compound  denominator 
may  be  converted  into  an  infinite  series  by  the  following  rule  : 
and  if  the  denominator  be  a  simple  quantity,  it  maybe  divided 
into  two  or  more  parts. 
25* 


282 


EXPANSION  OF  INFINITE  SERIES. 


RULE. 


Divide  the  numerator  by  the  denominator,  as -in  the  division 
of  integral  quantities,  and  the  operation  continued  as  far  as  may- 
be thought  necessary,  will  give  the  series  required. 

Ex.  1.  It  is  required  to  reduce into  an  infinite  series. 

a — aj 


1st  rem. 


2d  rem 


ax 

x^ 

a—x 

Quotient. 

a 

^2             <y>3            /*A 

x^ 
1                 

,1.  •  .  .  .   — 
a 

a?3     a?* 

a      a2 

4 

X 

^ 

X*      x^ 

fl2      a3 

J.- 

The  terms  in  the  quotient  are  found  thus ;  dividing  the 
first  remainder  rc^,  by  a,  the  first  term  of  the  divisor  a— a:, 

a;2 
we  shall  have  —  for  the  second  term  of  the  quotient,  because 

a 

the  division  can  be  only  indicated ;  multiplying  the  divisor  by 
— ,  and  subtracting  the  product  from  x"^,  the  remainder  is 

— ,  again,  dividing  this  remainder  by  a,  the  result  will  be  — , 

which  is  the  third  term  in  the  quotient ;  and,  in  like  manner, 
we  might  continue  the  operation  as  far  as  we  please  :  But  the 
law  of  continuation  is  evident,  because  the  powers  of  or  increase 
by  unity  in  each  successive  term  of  the  quotient,  and  the  powers 
of  a  increase  by  unity  in  the  denominator  of  each  of  the  terms 
after  the  first. 


EXPANSION  OF  INFINITE  SERIES.        283 

And  the  sum  of  the  terms  infinitely  continued  is  said  to  be 

equal  to  the  original  fraction .      Thus  we  say  that  thfe 

numerical  fraction  f ,  when  reduced  to  a  decimal,  is  equal  to 
.6666,  &c.,  continued  to  infinity. 


Ex.  2.  It  is  required  to  convert into  an  infinite  se- 


nes. 


a 

a — X 


X 


X" 
X 

a 


a- 


Quotient. 

,   ,  X  .  x"^  .  a?  .      . 


a     a''     a'' 


a 

a         a 


-2-,    &C.  • 

In  this  example,  if  x  be  less  than  a,  the  series  is  convergent, 
or  the  value  of  the  terms  continually  diipinishes ;  but,  when 
a;  is  greater  than  a,  it  is  said  to  diverge  :  Thus,  let  a=3  and 

a;=2,then  1+^+^+^+,  &c.  =1  +  2  +  4_f_^_|.^  &c. ; 

O/        Q,  CI 

where  the  fractions  or  terms  of  the  series  grow  less  and  less, 
and  the  farther  they  are' extended  the  more  they  converge  or 
approximate  to  0,  whichis  supposed  to  be  the  last  termor  limit. 

But  if  a=2,  and  a;=3,  then  1+-+^+^+,  &c.  =1-F 

a     a^      a^ 

|+f4-V  +  »  ^^-j  ^^  which  the  terms  become  larger  and 
larger.     This  is  called  a  diverging  series. 

Ex.  3.  It  is  required  to  convert  -— ; — into  an  infinite  series. 

1+a 


284        EXPANSION  OF  INFINITE  SERIES. 
1+a 
Quotient. 
1— a4-«^— «^+«*— o*4-«®~,  &c. 


1 
1+a 


— a 

— a — a^ 


a'^  +  a^ 
— a5,  &c. 

Whence  it  follows,  that  the  fraction •  is   equal  to  the 

l-j-a 

series,  l~'a-\-a^—a^-^a'^—a^-{-a^  —  a^-\-j  &c. 

372.  If  we  make  a=zl,  we  have  this  remarkable  compari- 
son :  —-— =1  —  1  + 1  —  1  +  1  —  1  +  1— j&c.  to  infinity;  which 
1+a 

appears  rather  contradictory;  for  if  we  stop  at  — 1,  the  se- 
ries gives  0  ;  and  if  we  finish  at  +1,  it  gives  +1.  The  real 
question,  however,  results  from  the  fractional  parts,  which 
(by  division)  is  always  +  J  when  the  sum  of  the  terms  is  0, 
and  —J  when  the  sum  is  +1  :  because  the  complete  quotient 
is  found  by  placing  the  remainder  over  the  divisor,  in  the  form 
of  a  fraction,  and  annexing  it  to  the  terms  in. the  quotient  with 
its  proper  sign  ;  but  the  remainder  in  the  present  case  is  + 1 , 
or  — 1  ;  hence  the  fraction  to  be  added  is  +^,  or  —J  ;  and, 
consequently,!  is  the  trae  quotient  in  the  former  case,  and 
1  — ^,  or  i  in  the  other.  This  will  appear  evident  by  taking 
successive  portions  of  the  series  ;  thus,  for  six  terms,  we  shall 
^ve  1  — 1  +  1  — 1-f  1  — l+i=r^,  and  for  seven  terms,  1  —  1 
+  l-l  +  l-l  +  l-i=f 

Scholium.  Here  we  might  infer,  by  conversion,  that  the 
sum  of  an  infinite  series  is  found,  \^en  we  know  the  fraction 
which  would  produce  such  a  series  by  actual  division  ;  but, 
although  it  is  a  fact  that  the  fraction  is  a  value  of  the  series, 
still  it  may  not  be  the  only  one  which  would  produce  the  same 
series  :  Thus,  the  above  series,  1 — 1  +  1  — 1  +  1  — 1  +  1  — l  +  i 
&c.,  to  infinity,  can  be  produced  by  several  other  fractions 
besides  the  fraction  ^. 


EXPANSION  OF  INFINITE  SERIES.        285 

Let,  for  example,  ^  be  converted  into  an  infinite  series  by 

actual  division  :    Now,  it  is  plain  that  4=,  .  ,  .  ,,  and  the 

^  ^     1  +  1  +  1* 

operation  will  stand  thus  : 


1 
1+1+1 

1+1+1 

Quotient. 

1-141— 1  +  1— l+.&c 

—1 

77 

+1 
+1+1 

+1        •    •  • 

-1-1 
_i— 1— 1 

+  1,&C. 

In  like  manner,  J  will  produce  the  above  series,  and  so  on. 

374.  Let  us  now  make  a^^;  and  the  preceding  develop- 
ment shall  be 


^f=i-i+i-i+A-A+.  &"•  = 


\+i 

The  sum  of  two  terms  is  J,  which  is  too  small  by  J  ;  three 
terms  give  -J,  which  is  too  much  by  -^  ;  for  the  sum  of  four 
terms,  we  have  |-,  which  is  too  small  by  gV?  ^^' 

We  see  here  that  the  successive  portions  of  the  series  are 
alternately  greater  and  less  than  the  fraction  f ,  which  repre- 
sent it ;  but  that  the  difference,  whether  it  be  in  excess  or 
deficiency,  becomes  less  and  less. 

375.  Suppose  again  a=:^,  and  we  shall  have 

Now,  by  considering  only  two  terms,  we  have  J,  which  is 
too  small  by  -f^  ;  three  terms  make  |-,  which  is  too  much  by 
■^Q  ;  four  terms  give  |^,  which  is  too  small  by  j^,  and  so  on. 

376.  The  fraction may  also  be  resolved  into  an  infi- 

1  +  a 

nite  series  another  way  ;  namely,  by  dividing  1  by  a+1,  as 
follows  : . 


286        EXPANSION  OF  INFINITE  SERIES. 


1 
1  + 


a+i 

Quot. 

1      l+i,_l  +  l_.&e. 


-9~r   _9 


It  is  however  unnecessary  to  carry  the  actual  division  any 
farther,  as  we  are  enabled  already  to  continue  the  series  to 
any  length,  from  the  law  which  may  be  observed  in  those 
terras  we  have  obtained  ;  the  signs  are  alternately  plus  and  mi- 

nuSy  and  each  term  is  equal  to  the  preceding  one  multiplied  by-. 

It  is  thus  by  changing  the  order  of  the  terms  of  the  deno- 
minator, we  obtain  the  quotienfunder  different  forms,  and  that 
we  pass  from  a  diverging  series,  for  certain  values  of  a,  to  a 
converging  series  for  the  same  values. 

It  may  also  be  here  observed,  that  in  the  division  of  the  two 
polynomials,  if  we  deviate  from  the  established  rule  (Art.  93), 
we  arrive  at  quotients  which  do  not  terminate  : 

Thus,  for  example,  a^—h"^,  divided  by  a-\-b,  according  to  the 
rule  above  quoted,  gives  for  the  quotient  a  —  b;  but  if  we  divide 
c^—.]P-  by  i-j-a.  we  shall  arrive  at  a  quotient  which  does  not 
terminate  :  thus, 


EXPANSION  OF  INFINITE  SERIES.        287 


62 


Here,  we  can  clearly  see  that  the  quotient  will  not  termi- 
nate, however  far  we  may  continue  the  operation,  because  we 
have  always  a  remainder. 

In  this  case,  by  taking  6 -fa  for  a  divisor,  we  must,  in  order 
to  find  the  quotient  a—h,  divide  the  whole  dividend  by  all  the 
divisor,  that  is  to  say,  a^ — b"^  or  [a-\-b)x(ci—h)  by  a-{-h. 

377.  When  there  are  more  than  two  terms  in  the  divisor, 
we  may  also  continue  the  division  to  infinity  in  the  same  man- 
ner. 

1 


Ex.  4.  It  is  required  to  convert :; ; — l    into  an  infinite 

\—a+a^ 


series. 
1 
l-a  +  a^ 


a — a^ 


— «3 


a^  +  a^ 


Quot. 

l+a  —  a^—a^+a^-i-a^j  &c. 


We  have  therefore 


a^ 

-a'  +  dS 

a7 -084.^9 

&c. 
1        _,^. 

288        EXPANSION  OF  INFINITE  SERIES, 
to   infinity:  where,  if  we  make  a=rl,  we  have 


1-1  +  1 

1  =  1  +  1  —  1  — 1  +  1  +  1,  &c.,  which  series  contains  twice  the 
series  found,  (Art.  372),  1  — 1  +  1  — l  +  l,  <fec.  Now,  as  we 
have  found  this  to  be  equal  to  i,  it  is  not  extraordinary  that 
we  should  find  |,  or  1,  for  the  value  of  that  which  we  have 
just  determined. 

By  making  a=^,we  shall  have  3  =  J  =  l+-t— i— J^+^ 

If  az=z^,  we  shall  have 

J=j=^+i-2j—iT^ih^  &c. 

9 

And  if  we  take  the  four  leading  terms  of  this  series,  we 
have  ^*,  which  is  only  jly  less  than  f. 

Let  us  suppose  again  a=:f,  and  we  shall  have  y  z=  |^= 1  +  J 

9" 

~~y?"  —  8T"^'AV"f"'  ^^'  ^^^^  series  is  therefore  equal  to  the 
preceding  one,  and  by  subtracting  one  from  the  other,  we  ob- 
tain ^ — Tf  ~8T "^TIjV'  *^^-»  which  is  necessarily  =0. 

378.  The  method  which  has  been  here  explained,  serves 
to  resolve,  generally,  all  fractions  into  infinite  series  ;  which 
is  often  found,  as  has  been  observed  by  Euler  in  his  Algebra^ 
to  be  of  the  greatest  utility ;  it  is  also  remarkable,  that  an  in- 
finite series,  though  it  never  ceases,  may  have  a  determiate 
value.  It  should  likewise  be  observed,  that  from  this  branch 
of  Mathematics,  inventions  of  the  utmost  importance  have 
been  derived,  on  which  account  the  subject"  deserves  to  be 
studied  with  the  greatest  attention. 

Ex.  5.  It  is  required  to  convert into  an  infinite  series 

a-{-x 

Ans.  1  -  -+-^ 3-+,  &c 

Ex.  6.  It  is  required  to  convert  — --r  into  an  infinite  series 

a+6 

c     .be   ,  b^c       Pc  ,      « 
Ans. ^+-r  — r  +  '  ^^ 

Ex.  7.  It  is  required  to  convert into  an  infinite  series 

.        b.        X  .  x^        x^   ,        ' 
Ans.  -(1 1— 2-  ~  -3-+»  ^^ 


BINOMIAL  THEOREM.  289 

Ex.  8.  It  is  required  to  convert into  an  infinite  series. 

a—x 

1  -\-x 

Ex  9.  It  is  required  to  convert into  an  infinite  series. 

\—x 

Ans.  l+2a:4-2a;34.2a;4  +  2a;5-f-,  &c. 

Ei.  10.  It  is  required  to  convert —  into  an  infinite  se- 

(a-f-x)2 

nes.  •  Ans.  1 1 — -+,  &c. 

a       a^       a^ 

Ex.  11.  It  is  required  to  convert  into  an  infinite  series. 

.        a      ax  .  ax^  .  ax^  ,      « 

Ans.  -4— i-H — r+-T+»  ^^' 
c       c^       c^       c* 

Ex.  12.  It  is  required  to  convert into  an  infinite  se- 

1        x\  x^      x^^     x^^ 
a^      a^     a'o     a^*     a^^ 

Ex.  13.  It  is  required  to  convert -,  or—- — -,  into  an  infi- 

fi        fi         fi         fi 
nit6  series.  Ans.  -+—  +__+_+,  &c. 

Ex.  14.  It  is  required  to  convert  -or- into  an  infinite 

4      5  —  1 

series. 

^  II.    INVESTIGATION  OF  THE  BINOMIAL  THEOREM. 

379.  Previous  to  the  investigation  of  the  Binomial  Theorem^ 
it  is  necessary  to  observe,  that  any  two  algebraic  expressions 
are  said  to  be  identical,  when  they  are  of  the  same  value,  for  all 
values  of  the  letters  of  which  they  are  composed.  Thus,  x — 1 
=a:  — 1,  is  an  identical  equation :  and  shows  that  x  is  indeter- 
minate ;  or  that  the  equation  will  be  satisfied  by  substituting, 
for  X,  any  quantity  whatever. 

Also,  {x-{-a)  X  [x—a)  and  x'^—a^,  are  identical  expressions  ; 
that  is,  (aj+a)  X  {x—-a)  =  x'^—a^;    whatever  numeral  values 
may  be  given  to  the  quantities  represented  by  x  and  a, 
26 


290  BINOMIAL  THEOREM. 

380.  When  the  two  members  of  any  identity  consist  of  the 
same  successive  powers  of  some  indefinite  quantity  x,  the  coeffi- 
cient of  all  the  like  powers  of  x,  in  that  identity^  will  be  equal 
to  each  other. 

For,  let  the  proposed  identity  consist  of  an  indefinite  num- 
ber of  terms  ;  as, 

a-\-bx-{-cx'^  +  dx^  +  &c.   =a'+b'x-^c'x^-\-d'x^+SLC. 

Then  since  it  will  hold  good,  whatever  may  be  the  value  of 
X,  let  x=:0,  and  we  shall  have,  from  the  vanishing  of  the  rest 
of  the  terms,  a=za\ 

Whence,  suppressing  these  two  terms,  as  being  equal  to 
each  other,  there  will  arise  the  new  identity  bx-^cx^-^-dx^-^ 
&c.  ==:b'x-{-c'x^-{-d'x^-\-SLc.  which,  by  dividing  each  of  its 
terms  by  x,  becomes 

b-}-cx-\-dx^j-&c.   =b'-{-c'x-[-d'x^-\'&c. 

And,  consequently,  if  this  be  treated  in  the  same  manner  as 
the  former,  by  taking  x=zO,  we  shall  have  bz=y,  and  so  on  ; 
the  same  mode  of  reasoning  giving  c=c^,  d=d\  &c.,  as  was 
to  be  shown. 

381.  Newton,  as  is  well  known,  left  no  demonstration  of 
this  celebrated  theorem,  but  appears,  as  has  already  been  ob- 
served, to  have  deduced  it  merely  from  an  induction  of  parti- 
cular cases,  and  though  no  doubt  can  be  entertained  of  its  truth 
from  its  having  been  found  to  succeed  in  all  the  instances  in 
which  it  has  been  applied,  yet,  agreeably  to  the  rigour  that 
ought  to  be  observed  in  the  establishment  of  every  mathemati- 
cal theory,  and  especially  in  a  fundamental  proposition  of  such 
generaluse  and  application,  it  is  necessary  that  as  regular  and 
strict  a  proof  should  be  given  of  it  as  the  nature  of  the  subject 
and  the  state  of  analysis  will  admit. 

382.  In  order  to  avoid  entering  into  a  too  prolix  investiga- 
tion of  the  simple  and  well-known  elements,  upon  which  the 
general  formulae  depends,  it  will  be  sufficient  to  observe,  that  it 
can  be  easily  shown,  from  some  of  the  first  and  most  common 
rules  of  Algebra,  that  whatever  may  be  the  operations  which 
the  index  (m)  directs  to  be  performed  upon  the  expression 
(a-f-a:)"',  whether  of  elevation,  division,  or  extraction  of  roots, 
the  terms  of  the  resulting  series  will  necessarily  arise,  by  the 
regular  integral  powers  of  x  ;  and  that  the  first  two  terms  of 
this  series  will  always  be  a'"-\-ma'"-'^x  ;  so  that  the  entire  ex- 
pansion of  it  may  be  represented  under  the  form 

a'"_|_;na'»-iir4  Ba'"-^a:2+Ca'"-3+ Da"'-'*a;3  +  &c. 
Where  B,  0,  D,  <fcc.  are    certain   numerical   coeflicients, 
that  are  independent  of  the  values  of  a  and  x  ;  which  two  lat- 
vcr  may  be  considered  as  denoting  any  quantities  whatever. 


BINOMIAL  THEORExM.  291 

383.  For  supposing  the  index  m  to  be  an  integer,  and  taking 
a=  I,  which  will  render  the  following  part  of  the  investigation 
more  simple,  and  equally  answer  the  purpose  intended ;  it  is 
plain  that  we  shall  have,  according  to  what  has  been  shown, 

{\-]-x)"'—l-\-mx+hx^-\-cx^-^dx^-\-,  &c (1) 

384.  And  if  the  index  m,  of  the  given  binomial,  be  negative, 
it  will  be  found  by  division,  that  (1  +  a;)— *",  or  the  equivalent 
expression 

\ = ! z=\'-mx—h'x'^—c'x'^—,  &c. 

(l-f-a;)™     \9\-mx-{-bx'^-\-cx'^,  &c. 

■where  the  law  of  the  terms,  in  each  of  these  cases  is  similar, 
to  that  above  mentioned. 

m 

385.  Again,  let  there  be  taken  the  binomial  (l-|-a:)'^>  hav- 

Tfl 

ing  the  fractional  index  —  ;  where  m  and  n  are  whole  positive 

gl^umbers. 

Then,  since  (l+a;)'"  is  the  nth  power  of  (l+a;)"  ;  and,  as 
above  shown,  {\-\'x)"'=:i-\-ax-\-b'^-{-cx^->rdx'^-\-,  &c.,  such  a 

series  must  be  assumed  for  (1-f  a;)"  ,.  that,  when  raised  to  the 
nth  power,  will  give  a  series  of  the  form  \-\-ax-\-hx'^-\-cx'^-\- 
dx^^,  &c. 

But  the  nth  or  any  other  integral  power  of  the  series  1  + 
px-\-qx'^--\-rx'^-\-sx^-\-,  &c.  will  be  found,  by  actual  multiplica- 
tion, to  give  a  series  of  the  form  here  mentioned  ;  whence^  in 
this  case,  also,  it  necessarily  follows,  that 

m 

(\-\-xY  z=\-[-px-\-qx'^-\-rx^-{-sx^-\-,  &c. 

And  if  each  side  of  this  last  expression  be  raised  to  the  nth 
power,  we  shall  have  {l-\-x)'"z=[l-\-{px-{-qx'^-^rx^-\-sx'^-\-, 
&,c.)]" ;  or,  by  actual  involution,  * 

l-{-mx-\-bx'^-\-cx^-{-,  Sic.  =:l-\-n{px-\-qx'^-{-,  (fee.)-}-,  &;c. 

Whence,  by  comparing  the  coefficients  of  x,  on  each  side 
of  this  last  equation,  we  shall  have,  according  to  (Art.  380), 

np=:m,  or  p=~  ;  so  that,  in  this  case, 
n 

(i+xY  =l-^'^x+qx^-\-rx^-\'Sx^'\-,  See (2); 

n 

where  the  coefficient  of  the  second  term,  and  the  several 
powers  of  x,  follow  the  same  law  as  in  the  case  of  integral 
powers 


292  BINOMIAL  THEOREM. 


386.  Lastly,  if  the  index  —  be  negative,  it  will  be  found 

n 

by  division  as  above,  that  (1  +  a;)  »"  or  the  equivalent  expres- 
sions, 

—^=——-} =  i_^a:-/^2-,  &c.  (3). 

(l-fa?)"       1-j — x-\-qx^,  &c. 

where  the  series  still  follows  the  same  law  as  before. 

387.  x\nd  as  the  several  cases,  (1,2,  3),  here  given,  are  of 
the  same  kind  with  those  that  are  designed  to  be  expressed  in 
universal  terms*,  by  the  general  formula  ;  it  is  in  vain,  as  far 
as  regards  the  first  two  terms,  and  the  general  form  of  the  se- 
ries, to  look  for  any  other  origin  of  them  than  what  may  be 
derived  from  these,  or  other  similar  operations. 

388.  Hence,  because  {a-{-!ic)"'=a"' (l-\ — V,  if  there  be  as- 
sumed {a-^-xyzzza""  -{■ma'^-^x  -j-  Bx'^  +  Cx^  +  T>x\  &c.  ;  or 
which  will  be  more  commodious,  and  equally  answer  the  de- 
sign proposed, 

0+-:)"=^+^;©+M^hM^)'+"^'= w- 

it  will  only  remain  to  determine  the  values  of  the  coefficients 
Aj ,  Aj,  A3,  &c.  and  to  show  the  law  of  their  dependence  on 
the  index  (m)  of  the  operation  by  which  they  are  produced. 

389.  For  this  purpose,  let  m  denote  any  number  whatever, 

whole  or  fractional,  positive  or  negative  ;  and  for  -,  in  the 

above  formula,  put  y-{-z  ;  then,  there  will  arise  ( IH — )'"  =  [1 

4- (y+j»)]'"=[(l+y)-f- ;?]'",  which  being  all  identical  expres- 
sions, when  taken  according  to  the  above  form,  will  evidently 
be  equal  to  each  other. 

390.  Whence,  as  the  numeral  coefficients,  Aj,  A^,  A3,  &c. 
of  the  developed  formulae,  will  not  change  for  any  value  that 
can  be  given  to  a  and  x,  provided  the  index  (m),  remains  the 
same,  the  two  latter  may  be  exhibited  under  the  forms 
[l+(y4-^)]'"=l  +  A,  {y+^)+A,  (y+^)2+,  &c.^ 

[(l+y)+^]'"  =  (l+y)"'+Ai^(14-y)'"-^  +  A202(l+y)m-2_|,&c. 

And,  consequently,  by  raising  the  several  terms  of  the  first 
of  these  series  to  their  proper  powers,  and  putting  l+y=|)in 
the  latter,  we  shall  have 

&c.  =p'»+Ajp'"-iz-f-Aap'«-2^2^A3i)«-3;23-f-,  &c. 


BINOMIAL  THEOREM. 


293 


l+A, 

A.y^  +  ^k^'^ 
A3y3  +  4Ay 


+  3A3y 
+  6Ay 

H-lOAgyS 
&c. 


;23+&C.  (5). 


V+    &C. 


391.  Or,  by  orderinglhe  terms,  so  that  those  which  are  af-| 
fectei  with  the  same  power  of  z  may  be  all  brought  together, 
and  arranged  under  the  same  head,  this  last  expression  will 
stand  thus : 

^2+ A3 
+  4A,y 
4- 10  Ay 
+  30A,y3 
&c. 

In  which  equation  it  is  evident,  that  both  y  and  z  are  inde- 
terminate, and  independent  of  the  values  Aj,  A^,  A3,  &c.  ; 
since  the  result  here  obtained  arises  solely  from  the  substitu- 
tion of  the  sum  of  these  quantities  for  -  in  equation  (4). 

392.  Hence,  as  the  first  terms  and  the  coefficients,  or  mul- 
tipl^rs  of  the  like  powers  of  z^  in  these  two  expressions,  are, 
in  this  case,  identical,  we  shall  have,  by  comparing  the  first 
column  of  the  left-hand  member  with  the  first  term  of  that  on 
the  right, 

H-Ajy  +  A2y24.A3y3+A4y*+  &:c.  =;?'", 
which  is  an  identity  that  verifies  itself;  since,  by  hypothesis, 
(l+y)'"=jo'",  and,  according  to  the  general  formula,  (l+y)*" 
=  1  +  A,y+A,y2+A3y3+  &c. 

393.  Also,  if  the  second  of  these  columns  be  compared  in 
like  manner,  with  the  second  on  the  right,  there  will  arise  the 
new  identity, 

Aj-f2A2y-f3A3y2-f.4A4y3=A,p"»-i;  which  will  be  suffi- 
cient, independently  of  the  rest  of  the  terms  for  determining  the 
values  of  the  coefficients  A^,  Aj,  A3,  &;c. 

o"*        A 
For  since  k^f^-^^k^—=:—^  (1  +  k^y  +  k^'^-\-k^^ 

&;c.),  the  equating  this   series  with  the  last,  and  multiplying 
the  left-hand  side  by  1+y,  will  give 

■[Aj  +  2A,y+3A3y2-f&;c.]  (1  +  y)=A^+AjA,3/+AjA,y24.A, 
k^"^  -f  &c. 

And,  therefore,  by  actually  performing  the  operation,  and 
arranging  the  terms  accordingly,  we  shall  have 


A,+2A, 


y+3A3 

+  2A, 


y2+4A, 
+  3  A3 


y3+  &c. 


=  Aj4-A,A,y  +  A,A,y2+A,A3y3+  &c. 
394.  From  which  last  identity,  there  will  obviously  arise, 
by  equating  the  homologous  terms  of  its  two  members,  the  fol- 
lowing relations  of  the  coefficients  : 
26* 


294 


BINOMIAL  THEOREM. 


\        A,=A, 
2A2=A,Aj— Aj 


uAg — AjA2"^2A2 
4A^=AjA3— 3A3 


nA  =AjA  _-("-!) A 


or 


A,: 
A,= 


A,=A. 
A.(A.-1) 

2 
A,(A,-2) 


A3(A,-3) 


A_-[A-(»-l)) 


And,  consequently,  as  the  coefficient  Aj  of  the  second  term 
of  the  expanded  binomial,  has  been  shown  to  be  equal,  in  all 
cases,  to  the  index  {m)  of  the  proposed  binomial,  the  last  of 
these  expressions  will  become  of  the  form 
Aj=:m 

m{m  —  1)  • 

^^=—2 

_m{m  —  l).{m—2) 

_m(m  — l).(m— 2).(m  — 3) 

^*-         2":3:4 

m{m—l).{m  —  2).(m—3)  ....   [m—{n—l)] 

""  2.3.4.5 7~^  ' 

where  the  law  of  the  continuation  of  the  terms,  from  A^  to  the 
general  term  A„,  is  sufficiently  evident. 

395.  Whence  it  follows,  that,  whether  the  index  m  be  in- 
tegral or  fractional,  positive  or  negative,  the  proposed  binomial 
(a+a:)'",  when  expanded,  may  always  be  exhibited  under  the 
form 

a"'(l'\--Y= 
&c.]; 


or  (a-f  3?)'"  = 


<f'\-ma''^^x-\- 


m(m—l)  m{m—l)  {m—2) 


x-{ 


2.3. 


^m_3^3  ^c. 


And  if be  substituted  in  the  place  of  H — ,  the  same  for- 

a  a 

mula  will,  in  that  case,  be  expressed  as  follows  : 


BINOMIAL  THEOREM.  295 

m{m — l)/ar> 


,          .                          ,     ,  m(m  —  1)        -  _ 
or  la — j:)'"  =  a'" — ma^—^x-t—^—x a"*— ^a;^ — 

2.4 
Where  it  is  to  be  observed,  that  the  series,  in  each  of  these 
cases,  will  terminate  at  the  (m+l)th  term,  when  m  is  a  whole 
positive  number  ;  but  if  m  be  fraction|4|or  negative,  it  will 
proceed  ad  infinitum  ;  as  neither  the  facrors  m— 1,  m — 2,  m— 
3,  &c.  can  then  become  =rO. 

396.  To  this  we  may  add,  that  in  the  two  last  instances 

here  mentioned,  the  second  term  ( -)  of  the  binomial  must  be 

less  than  1,  or  otherwise  the  series,  after  a  certain  number  of 
terms,  will  diverge,  instead  of  converging. 

397.  It  may  also  be  farther  remarked,  that  when  a  and  x 
in  these  formulae,  are  each  equal  to  1,  we  shall  have,  agree- 
ably to  such  a  substitution,  (a4-n)"'=(l  +  l)'"=2'"  =  l+7n-f- 
m{m—\)     m(m-'\)  .  {m—2)     m{m  —  l)  .  (m— 2)  .  (m— 3) 

2        "^  2^3  '  2.3.4  '        * 

&c.,  and 

(a—x)"'=.{l  —  l)'^  =  0'"  =  0—l—m-\' 

m{m—l)  __  m(m— 1)  .  (m—2)     m{m^l)  .  (m— 2)  .  (m—3) 

2  2^3  '  2.3.4 

— ,  &c. 

From  which  it  appears,  that  the  sum  of  the  coefficients 
arising  out  of  the  development  of  the  mth  power,  or  root  of 
any  binomial,  is  equal  to  2'" ;  and  that  the  sum  of  the  coeffi- 
cients of  the  odd  terms  of  the  mth  power,  or  root  of  a  resi- 
dual quantity,  is  equal  to  the  sum  of  the  coefficients  of  the 
even  terms. 

m  0  0—1 

398.  Finally,  let  m=0  ;  then  (a-|-a;)=a4'0Xa      x  -{• 

?(t±V"V4-,  &c.,  =a+0  .  --f-0  .  ^+,  &c. 
2  a  a^ 

where  it  is  evident  that  the  series  terminates  at  the  first  term 
(a°)i  since  the  coefficient  of  every  successive  term  involves 
0  for  one  of  its  factors  ;  therefore  {a-{-xy=a^=l,  (Art.  86). 
And,  if  a=a: ;  then  (a-^x)^=a^=l,  that  is,  0°=1.     Hence,  it 


296  BINOMIAL  THEOREM. 

follows,  that  any  quantity,  either  simple  or  compound,  raised 
to  the  power  0,  is  equal  to  unity  or  1  ;  and  also  that  0^  is  in 
all  cases  equal  to  unity  or  1 . 

399.  Although  it  has  been  observed,  that  O*'  appears  to 

admit  of  an  infinity  of  numerical  values  ;  because  it  is  equal 

to  ■§-,  which  is  the  mark  of  indetermination  ;  yet  it  is  plain, 

from  what  is  above  shown,  that  0'^  is  only  one  of  the  values  of 

0™  0 

g,  which,  in  that  particular  case,  where  — =0*^=:-,  is   equal 

to  unity.  The  intelligent  reader  is  referred  to  Bonnycastle's 
Algebra,  8vo.  vol.  i^  Also,  Lagrange's  Theorie  des  FonC' 
tions  Analytiques^  aWrLecons  sur  le  Calcul  des  Fonctions. 


§  III.    APPLICATION  OF  THE  BINOMIAL  THEOREM  TO  THE 
EXPANSION  OF  SERIES. 

400.  The  method  of  expanding  any  binomial  of  the  form 
(a^^xY^  when  m  is  any  whole  number  whatever,  has  been 
already  pointed  out,  and  it  has  also  been  observed,  that  the 
series  will  always  terminate,  when  m  is  a  whole  number  : 
But  when  m  is  a  negative  number,  or  k  fraction,  then  the  se- 
ries expressing  the  value  of  (a-\-xY  does  not  terminate. 

Let  m='^,  and  substitute  •"  for  m  in  the  series  then 


»■   ,  nor 

=za   H 

r 


jiifi ri/ofi\ 

+  ^  ^  ^  -(  -g-j  +  J  <&^c.,  which  is  a  general  expression  for  find- 

» 

ing  the  approximate  value  of  any  binomial  surd  quantity,  •■ 
being  either  positive  or  negative,  n  and  r  any  whole  numbers 
whatever. 

Ex.  1.  Find  the  approximate  value  of  3 ^  (i^+c^)  or  (i^-l-c^)^. 


Here  a=^h^ 

n=l 
r=3 


7W""3\P/      363' 
n{n-\)(x\_\{-^)(c^\_ 


3r2 


W/     2.32  w)~'    32^8' 


BINOMIAL  THEOREM. 


297 


¥.3r^        ~Wy~        2.3.33       W)~3^l 


5c» 
69 


&c.  =  &c. 


-3  y.6  5^9 

1 


Ex.  2.  Find  the  value  of 


(b+cf 


&c.). 
or  (64-c)~^in  a  series. 


Here  a=b~ 
c=x 


71: 


1  1 


nix 


2c 


(D=-f= 


J 


2r 


^__2_l)/c2> 
2 

<fec.=:&C. 


3c2 


WJ~  2  VPJ-'62' 


1  1  /,      2c   ,  3c2     4c3  .    „      \  • 

Hence  ^^-j_^=--(l_-^4-^-_-  +  &c.) 

-  1 

401.  Now  let  71  =  1,  (a  +  a:)'-=(a+a;)'-=;^(a-f-a;);    and  a 

n 

~=y  a  ;  hence  the  series  (Art.  398)  is  transformed  into  {/ 

-h&c.) (A). 

Let  a=  1,  a;  =  l  ;  then  {/ 2  =  1  + 
+  &C (B). 


■1     1-r     (l-r).(l^r) 
2r2  ~^       2  .  3  .  r3 

Thus,  if  .=2.  then   ^2  =  1 +^-^3+^-1+^-1^ 

+  &C.     And  if  r=3, 

.    «     ,   .  1      1       5       2.5    ,  2.11    ,  2.7.11 
then^3  =  l  +  3-3,+  --3,-+-^+-3^+  &c 

By  means  of  the  series  marked  A,  the  rth  root  of  many  other 
numbers  may  be  found  ;  if  a  and  x  be  so  assumed,  that  a?  is  a 
small  number  with  respect  to  a,  and-(/  a,  a  whole  number. 

Ex.  3.  It  is  required  to  convert  -y/S,  or  its  equal  y'(44-l), 
into  an  infinite  series. 

Here  a  =  4,  x=zl,  r=2;  then{/ a=^y^4=r2,  and  we  have 

V(4+l)=2(l  +  L_i_+i._^±+&c.) 

Ex.  4.  It  is  required  to  convert  ^  9,  or  its  equal  ^z  (8+ 1) 
into  an  infinite  series. 


298  BINOMIAL  THEOREM. 

Here  a— 8,  x=zl,  r=3  ;  ihen{^  a=^  8=2,  and  we  obtain 
/  .8+1)=^  9=2  I  1  +  3L_  J_+-i__|lH.&e. 

402.  The  several  terms  of  these  series  are  found  by  sub- 
stituting for  a,  X,  and  r,  their  values  in  the  general  series  mark- 
ed (A)  or  (B),  and  then  rejecting  the  factors  common  to  both 
the  numerators  and  denominators  of  the  fractions. 

Thus,  for  instance,  to  lind  the  5th  term  of  the  series  express- 
ing the  approximate  value  of -J/  9,  we  take  the  5th  term  of  the 
general  series  marked  (A),  which  is 

.-.•the  value  of  the  fraction  is  -^^^^i  ^,  )  =  -    ^3^3.33 

2.5  2.5^^. 

=—^04  Q  Q3~ — 05 — oT'     ^"  ^"^^  manner  each  term  of  the 

series  is  calculated ;  and  the  law  which  they  observe  is,  that 
the  numerators  of  the  fractions,  consist  of  certain  combinations 
of  prime  numbers,  and  the  denominators  of  combinations  01  cer- 
tain powers  of  a  and  r. 

3  . 

Ex.  5.  Find  the  value  of  (c^ — a:^)*  in  a  series. 

.  .  -/,        3a:2         3a:*         5x^       ^      \ 

Ex.  6.  It  is  required  to  convert  ^  6,  or  its  equal  3/  (8—2), 
into  an  infinite  series. 

Ex.  7.  It  is  required  to  extract  the  square  root  of  10,  in  an 
infinite  series.  Ans.  3+ -^i^  --l-^+^l±^,-&o. 

Ex.  8.  To  expand  a'^{a^—x)  ^  in  a  series. 

Ex.  9.  To  find  the  value  of  ^  (a^-^-x^)  in  a  series. 

.    x^       2a:io        6j;^5 

Ans.  a-1 —  <Scc. 

/^"^-  ''^Sa*      25a»^125ai* 

Ex.  10.  Fnd  the  cube  root  of  1  — x^,  in  a  series. 

,      a;3      x^       5x^      \0x^^       , 


CHAPTER  XIII. 


ON 


PROPORTION    AND    PROGRESSION. 

§  I.    ARITHMETICAL   PROPORTIOx\  AND   PROGRESSION. 

403.  Arithmetical  Proportion  is  the  relation  which  two 
numbers,  or  quantities,  of  the  same  kind,  have  to  two  others, 
when  the  difference  of  the  first  pair  is  equal  to  that  of  the  se- 
cond. 

404.  Hence,  three  quantities  are  in  arithmetical  proportion, 
when  the  difference  of  the  first  and  second  is  equal  to  the  dif- 
ference of  the  second  and  third.  Thus,  2,  4,  6  ;  and  a,  a+b, 
a -{-2b,  are  quantities  in  arithmetical  proportion. 

405.  And  four  quantities  are  in  arithmetical  proportion, 
when  the  difference  of  the  first  and  second  is  equal  to  the 
difference  of  the  third  and  fourth.  Thus,  3,  7,  12,  16  ;  and 
a,  a-f-^j  c,  c-{-b,  are  quantities  in  arithmetical  proportion. 

406.  Arithmetical  Progression  is,  when  a  series  of 
numbers  or  quantities  increase  or  decrease  by  the  same  com- 
mon difference.  Thus  1 ,  3,  5,  7,  9,  &c.  and  a,  a-\-d,  a-\-2d^ 
a+3t?,  &c.  are  an  increasing  series  in  arithmetical  progres- 
sion, the  common  differences  of  which  are  2  and  d.  And  15, 
12,  9,  6,  &LC.  and  a,  a — d,  a— '2d,  a  —  Sd,  &;c.  are  decreasing 
series  in  arithmetical  progression,  the  common  differences  of 
which  are  3  and  d. 

407.  It  may  be  observed,  that  Garnier,  and  other  Euro- 
pean writers  on  Algebra,  at  present,  treat  of  arithmetical  pro- 
portion and  progression  under  the  denomination  of  equi-differ- 
ences,  which  they  consider,  as  Bonnycastle  justly  observes, 
not  without  reason,  as  a  more  appropriate  appellation  than  the 
former,  as  the  term  arithmetical  conveys  no  idea  of  the  nature 
of  the  subject  to  which  it  is  applied. 

408.  They  also  represent  the  relations  of  these  quantities 
under  the  form  of  an  equation,  instead  of  by  points,  as  is  usu- 
ally done  ;  so  that  if  a,  b,  c,  d,  taken  in  the  order  in  which 
they  stand,  be  four  quantities  in  arithmetical  proportion,  this 


300        PROPORTION  AND  PROGRESSION. 

relation  will  be  expressed  by  0—6  =  c — d\  where  it  is  evi- 
dent that  all  the  properties  of  this  kind  of  proportion  can  be 
obtained  by  the  mere  transposition  of  the  terms  of  the  equa- 
tion. 

409.  Thus,  by  transposition,  a-\-d=h-\-c.  From  which  it 
appears,  that  the  sum  of  the  two  extremes  is  equal  to  the  sum  of 
the  two  means  :  And  if  the  third  term  in  this  case  be  the  same 
as  the  second,  or  c  =  &,  the  equi-diflference  is  said  to  be  con- 
tinued, and  we  have 

a-^d—2b  ;  or  bz=z\(a-ird)  ; 
where  it  is  evident,  that  the  sum  of  the  extremes  is  double  the 
mean  ;  or  the  mean  equal  to  half  the  -sum  of  the  extremes. 

410.  In  like  manner,  by  transposing  all  the  terms  of  the 
original  equation,  a — ^  =  c — <f,  we  shall  have  b—a=d—c; 
which  shows  that  the  consequents  b,  d,  can  be  put  in  the 
places  of  the  antecedents  a,  c ;  or,  conversely,  a  and  c  in  the 
places  of  b  and  d. 

411.  Also,  from  the  same  equality  a  —  bz=:c—d,  there  will 
arise,  by  adding  m—n  to  each  of  its  sides, 

(a+m) — {b-}'n)z=[c-\-m)  —  {d-\-n)  ; 
where  it  appears  that  the  proportion  is  not  altered,  by  aug- 
menting the  antecedents  a  and  c  by  the  same  quantity  m,  and 
the  consequents  b  and  d  by  another  quantity  n.  In  short, 
every  operation  by  way  of  addition,  subtraction,  multiplica- 
tion, and  division,  made  upon  each  member  of  the  equation, 
a — b=c  —  d,  gives  a  new  property  of  this  kind  of  proportion, 
without  changing  its  nature. 

412.  The  same  principles  are  also  equally  applicable  to 
any  continued  set  of  equi-differences  of  the  form  a — b^=b  — 
cz=c—d=d—€y  &c.  which  denote  the  relations  of  a  series  of 
terms  in  what  has  been  usually  called  arithmetical  progres- 
sion. • 

413.  But  these  relations  will  be  more  commodiously  shown, 
by  taking  a,  b,  c,  d,  (fee.  so  that  each  of  them  shall  be  greater 
or  less  than  that  which  precedes  it  by  some  quantity  d^ ;  in 
which  case  the  terms  of  the  series  will  become 

a,  a±r/'',  a±2J^  a±3d',  a±46?',  (fee. 
Where,  if  /  be  put  for  that  term  in  the  progression  of  which 
the  rank  is  n,  its  value,  according  to  the  law  here  pointed  out, 
will  evidently  be 

/=a±(n-lK; 
which  expression  is  usually  called  the  general  term  of  the  se- 


PROPORTION  AND  PROGRESSION.         301 

ries  ;  because,  if  1,  2,  3,  4,  &c.  be  successively  substituted 
for  n,  the  results  will  give  the  rest  of  the  terms. 

Hence  the  last  term  of  any  arithmetical  series  is  equal  to  the 
first  term  plus  or  minus,  the  product  of  the  common  difference^ 
by  the  number  of  terms  less  one. 

414.  Also,  if  s  be  put  equal  to  the  sum  of  any  number  of 
terms  of  this,  progression,  we  shall  have 

^=a  +  (airf')  +  (a±2(/)-f    ....   ■\-[a^(n-\)d'l 
And  by  reversing  the  order  of  the  terms  of  the  series, 
s  =  [aJc[n-l)d']^[aJ^{n—2)d']-ir   •  •  •  {a±d')-ira. 
Whence,  by  adding  the  corresponding  terms  of  these  two 
equations  together,  there  will  arise 
,    2s=:.\2a±{n—\)d''\-{-[2a^{n  —  \)d''\,  &c.  to  n  terms. 
And,  consequently,  as  all  the  n  terms  of  this  series  are  equal 
to  each  other,  we  shall-have 

2sz=:n[2a^[n—l)d%oxs-l[2a^{n-\)d']  .   .  (1). 

415.  Or,  by  substituting  I  for  the  last  term  a±(n  — l)(i',  as 
found  above,  this  expression  (1)  will  become 

s:=l{a  +  l)     .     .     .     .     (2).^ 

Hence,  the  sum  of  any  series  of  quantities  in  arithmetical 
progression  is  equal  to  the  sum  of  the  •  two  extremes  multiplied 
by  half  the  number  of  terins. 

It  may  be  observed,  that  from  equations  (1)  and  (2),  if  any 
three  of  the  five  quantities,^,  d\  n,l,  s,  be  given,  the  rest  may 
be  found. 

416.  Let  /,  as  before,  be  the  last  term  of  an  arithmetic  se- 
ries, whose  ^r^^  term  is  (a),  common  difference  {d')y  and  num' 

%er  of  terms  in) :  then  Z=a-f  (n— l)dt^ ;  .*.  <i^= -.        Now 

^  ^  n  —  1 

the  intermediate  terms  between  the  first  and  the  last  is  n — 2 ; 

let  «— 2=:m,then   n  — l=7n+l.     Hence,  c^'= — —-,  which 

m-f-l 
gives  the  following  rule  for  finding  any  number  of  arithmetic 
means  between  two  numbers.  Divide  the  difference  of  the  two 
numbers  by  the  given  number  of  means  increased  by  unity,  and 
the  quotient  will  be  the  common  difference.  Having  the  com- 
mon difference,  the  means  themselves  will  be  known. 

Example  1.  Find  the  sum  of  the  series  1,  3,  5,  7,  9,  11, 
&c.  continued  to  120  terms. 

^'''d'=2'  \  •■•  -^P^+C'^  -  l)d']}.='¥\?Xl+{\20^ 
»=120'5l)2]=14400. 
27 


302        PROPORTION  AND  PROGRESSION. 


Ex.  2.  The  sum  of  an  arithmetic  series  is  567,  i\\(i first  tefm 
7,  and  the  common  difference  2.    What  are  the  number  of  terms  ? 
Here  ^=567,  \  /.  2s=n[2a+{n  —  \)d]:=^n[lA-^(n  —  1)2] 
a=7,      [  =l4n-\-2n'^—  2n=.1134  ;  .-.  n'^-\-6n-\-9  = 
d'=2;     )576,  and  n=:21. 
Ex.  3.  The  .ywm  of  an  arithmetic   series  is  1455,  the  first 
term  5,  and  the  number  of  terms  30.     What  is  the  common  dif- 
ference? Ans.  3. 
Ex.  4.  The  sum  of  an   arithmetic  series  is  1240,  common 
difference  4,  and  number  of  terms  20.     What  is  the  first  term? 

Ans.  100. 

Ex.  5.  Find  the  sum  of  36  terms  of  the  series,  40,  38,  36, 

34,  &c.  Ans.  108. 

Ex.  6.  The  sum  of  an  arithmetic  series  is  AAO,  first  term  3, 

and  common  difference  2.     What  are  4,he  number  of  terms  ? 

Ans.  20. 

Ex.  7.  A  person  bought  47  sheep,  and  gave  1  shilling  for 

the  first  sheep,  3  for  the  second,  5  for  the  third,  and  so  on. 

What  did  all  the  sheep  cost  him?  Ans.  110/.  9s. 

Ex.  8.  Find  six  arithmetic  means  between  1  and  43. 

Ans.  7,  13,  19,25,  31,  37. 

§  II.    GEOMETRICAL  PROPORTION  AND  PROGRESSION. 

417.  Geometrical  Proportjpn,  is  the  relation  which  two 
numbers,  or  quantities,  of  the  same  kind,  have  to  two  others, 
when  the  antecedents  or  leading  terms  of  each  pair,  are  the 
same  parts  of  their  consequents,  or  the  consequents  of  their 
antecedents. 

418.  And  if  two  quantities  only  are  to  be  compared  togethep, 
the  part,  or  parts,  which  the  antecedent  is  of  the  consequent, 
or  the  consequent  of  the  antecedent,  is  called  the  ratio ;  ob- 
serving, in  both  cases,  to  follow  the  same  method. 

419.  Direct  proportion, '\s  when  the  same  relation  subsists 
between  the  first  of  four  quantities,  and  the  second,  as  between 
the  third  and  fourth. 

Thus,  a,  ar,  b,  br,  as  in  direct  proportion. 

420.  Inverse,  or  reciprocal  proportion,  is  when  the  first  and 
second  of  four  quantities  are  directly  proportional  to  the  re- 
ciprocals of  the  third  and  fourth. 

Thus,  a,  ar,  br,  b,  are  inversely  proportional ;  because  a,  ar 

^r-,  -r,  are  directly  proportional. 
or    0 

421.  The  same  reason  that  induced  the  writers  mentioned 


PROPORTION  AND  PROGRESSION.         303 

in  (Art.  407),  to  give  the  name  of  equi-differences  to  arithmeti- 
cal proportionals,  also  led  them  to  apply  that  of  equi-quotients 
to  geometrical  proportionals,  and  to  express  their  relations  in 
a  similar  way  by  means  of  equations. 

Thus,  if  there  be  taken  any  four  proportionals,  «,  b,  c,  d, 
which  it  has  been  usual  to  express  by  means  of  points,  as 
below, 

a  :  b  :  :  c  :  d. 

This  relation,  according  to  the  method  above-mentioned, 

will  be  denoted  by  the  equation  ^=-7,  (Art.  24);  where  the 

equal  ratios  are  represented  by  fractions,  the  numerators  of 
which  are  the  antecedents,  and  the  denominators  the  conse- 
quents.    Hence,  ad~bc. 

422.  And  if  the  third  term  c,  in  this  case,  be  the  same  as' 
the  second,  or  c=b,  the  proportion  is  said  to  be  continued* 
and  we  have  ad=b^,  b  =  '\/ad;  where  it  is  evident,  that  the 
product  of  the  extremes  of  three  proportionals,  is  equal  to  the 
square  of  the  mean  :  or,  that  the  mean  is  equal  to  the  square  root 
of  the  product  of  the  two  extremes. 

423.  Also,  from  the  equality,  Y=  J,  there  will  result  — r — 

0     a  0 

=  —7-  :  for,  by  adding  or  subtracting  1  from  each  side  of  the 

,        a  c  a-X-b     c-X-d        , 

equation;  then  TdLl  =  T±l  ;  •••-t^  =  -^-»  and  a^b  :b  :  : 
0.  a  0  d 

c^d  :  d. 

Hence,  when  four  quantities  are  proportionals,  the  sum  or  dif- 
ference of  the  first  and  second  is  to  the  second  as  the  sum  or  dif- 
ference of  the  third  and  fourth,  is  to  the  fourth. 

424.  In  like  manner,  if  a  :  6  :  :  c  :  (^ ;  then,  ma  :  mb  :  :  ^c  : 

b 

Hence,  when  four  quantities  are  proportionals,  if  the  first  and 
second  be  multiplied,  or  divided  by  any  quantity,  and  also  the 
second  and  fourth,  the  resulting  quantities  will  still  be  propor- 
tionals. 

a     c  a"       c" 

425.  Also,  U  a  :  b  :  :  c  :  d;  then  y=^  ;  .-.  j—=~-,  and  a" : 

0     a  0       a" 

6"  :  :  c"  :  t/" ;  where  n  may  be  any  number  either  integral  or 
fractional. 


\d.     Forv=^  ;  .-.(Art.  118),  ~=^;  and,  ma  :  mb  :  :  ^c 


304  PROPORTION  AND  PROGRESSION. 

Hence,  if  four  quantities  he  proportionals^  any  power  or  root 
of  those  quantities  loill  be  proportionals. 

And,  by  proceeding  in  a  similar  manner,  all  the  properties 

and  transformations  of  ratios  and  proportion^  can  be  easily  ob- 

a     c 
tained  from  the  equality -=3,  or  adz=bc. 
0     a 

426.  In  addition  to  what  is  here  said,  it  may  be  observ- 
ed, that  the  ratio  of  two  squares  is  frequently  called  duplicate 
ratio  ;  of  two  square  roots,  suoduplicate  ratio ;  of  two  cubes, 
triplicate  ratio ;  arid  of  two  cube  roots,  suhtriplicate  ratio. 
See  the  Appendix  at  the  end  of  this  Treatise,  where  the  doc- 
trine of  ratios  and  proportion  is  fully  explained  and  clearly 
illustrated. 

.  427.  Geometrical  Progression,  is  when  a  series  of  num- 
bers, or  quantities,  hava  the  same  constant  ratio,  or  which  in- 
crease, or  decrease,  by  a  common  multiplier,  or  divisor.  Thus, 
the  numbers  1,  2,  4,  8,  16,  &c.  (which  increase  by  the  continual 
multiplication  of  2),  and  the  numbers  1,  i,  i,  J^,  &c.  (which 
decrease  by  the  continued  division  of  3,  or  multiplication  of  i), 
are  in  Geometrical  Proofression. 

428.  In  general,  if  a  represents  thej^r^^  term  of  such  a 
series,  and  r  the  common  multiplier  or  ratio ;  then  may  the 
series  itself  be  represented  by  a,  ar,  ar^,  ar^,  ar^,  &:c.,  which 
will  evidently  be  an  increasing  or  decreasing  series,  according 
as  r  is  3.  whole  number,  or  a,  proper  fraction.  In  the  foregoing 
series,  the  index  ofr  in  any  term  is  less  by  unity  than  the  num- 
ber which  denotes  the  place  of  that  term  in  th§  series.  Hence, 
if  the  number  of  terms  in  the  series  be  denoted  by  (n),  the  last 
term  will  be  ar"-^ 

429.  Let  I  be  the  last  term  of  a  geometric  series,  then  /= 

/  «-i  /  / 

ar*-^  and  r''-^=—;   ..  r=      /-      The    number  of   interme- 
a  'SJ  a 

diate  terms  between  the  first  and  last  is  n— 2  ;  let  n-^2=m, 

«.+!  n 
then  71— -1=^+1,  and  r=:       A-,  which  gives  the   following 

rule  for  finding  any  number  of  geometric  means  between  two 
iSfumbers ;  viz.  Divide  one  number  by  the  other,  and  take  that 
root  of  the  quotient  which  is  denoted  by  m-\-l  ;  the  result  will 
he  the  common  ratio.  Having  the  common  ratio,  the  means  are 
found  by  multiplication. 

430.  Let  S  be  made  to  denote  the  sum  of  n  terms  of  the 
series,  including  the  first,  then 

a-Jtar+ar^+ar^-^ -f  a7--2+af-  i  =  S 


PROPORTION  AND  PROGRESSION.        305 

Multiply  the  equation  by  r,  and  it  becomes 
ar-{-ar^-{-ar^-\- -{-ar"—^=ar'*—^-\-af=rS. 

Whence,  subtracting  the  first  of  these  equations  from  the 
second,  observing  that  all  the  terms  except  a  and  ar"  destroy 
each  other,  we  shall  have 

ar"-a=rS-S=:(r-l)S;  and  .-.  S=^^^      .     .     .      (1). 

Or,  by  substituting  I  for  the  last  terra  ar"-^,  as  above  found, 

this  expression  will  become   S=: ;    from    which    two 

equations,  if  any  three  of  the  quantities  a,  r,  n,  I,  S,  be  given, 
the  rest  may  be  found.     Thus,  from   the  sedbnd   equation, 

7     /       INC           S-a        ,  ,     (r~l)S+« 
a=rl—{r—l)S  ;  r=- — -,  and  1=^ . 

In  the  formula   (1),  when  r=\,   we  have  S=- — r=7:. 

Now,  the  value  of  the  symbol  -,  in  this  particular  case,  shall 

be  equal  to  na  ;  because  the  series  a-\- ar -\- ar^ -\-  .... 
ffr"— ^-f-ar"— 1,  for  r=l,  becomes  a  +  «+«+«+>  <^c.,  and  the 
sum  of  n  terms  of  this  series,  is  evidently  equal  to  na  ;  there- 

^0  ^        .         or"— a  r"— 1  1— r" 

fore  S=-=:n«.      Or,  since  ;— — a  • r— «X— —  = 

0  r— 1  r— 1  1— r 

a,  [rn-i  +  r'^-z+r'—^..  +r+ l]=;tf  X  [l+r+r2-}«r3  .  .  ^n— i]^ 

which,  in  the  case  of  7  =  1,  becomes  a.  [i  +  l  +  l  +  j  &c.],and 
the  sum  of  w  terms  of  the  series  l  +  l  +  l-f-j  <fec.  is  evidently 

1  — r"  0 

equal  to  n  ;  therefore  S=ia  . z=za  .-  =  «.  (l-fl  +  l-f, 

&c.)  —aXn-=an,  as  before. 

431.  When  the  common  factor  r,  in  the  above  series,  is  a 
whole  number,  the  terms  a,  ar,  ar^,  ar'^—^,  form  an  increasing 
progression  ;  in  which  case  n  may  be  so  taken,  that  the  value 
of  the  sum  (S)  shall  be  greater  than  any  assignable  quantity. 

432.  But  if  r  be  a  proper  fraction,  as  -„  the  series  a,  „  -7^, 
— ,  will  be  a  decreasing  one,  and  the  expression  (Art.  430), 

by  substituting  ->  for  r,  and  changing  the  signs  of  the  numera- 

7* 


306        PROPORTION  AND  PROGRESSION. 

tor  and  denominator,  will  become    — ^ — - — -  ;  where  it  is 

r  — 1 

plain,  that  the  term  -j^  will  be  indefinitely  small  when  n  is 
incTefinitely  great  ;  and  consequently,  by  prolonging  the  se- 
ries,  S  may  be  made  to  differ  from  — — -  by  less  than  any  as- 
signable quantity. 

433.  Whence,  supposing  the  series  to  be  continued  indefi- 

nitely,  or  without  end,  we  shall  have  in  that  case,  Sr=— — -  ; 

which  last  expression  is  what  some  call  the  radix,  and  others 
the  limit  of  the  series  ;  as  being  of  such  a  value,  that  the  sum 
of  any  number  of  its  terms,  however  great,  can  never  exceed 
it,  and  yet  may  be  made  to  approach  nearer  to  it  than  by  any 
given  difference. 

434.  If  the  ratio,  or  multiplier,  r,  be   negative,  in  which 
case  the  series  will  be  of  the  form 

a—ar-\-ar^ — ar^-\- _|-ar"—'^,  where  the  terms 

are  -f  and  —  alternately,  we,  shall  have  S  = — -z — . 

And  if  r  be  a  proper  fraction,  -,  as  before,  we  shall  have, 

for  the  sun^f  an  indefinite  number  of  terms  of  the  series  a— 
a  ,    a         a    ,      .        _,       ar" 


Ex.  1.  Find  the  sum  of  the  series,  1,  3,  9,  27,  &c.  to  12 
terms. 

Here  a=l,x  ar"  -  a  _lx3^^ — 1  _8V-' I 

r=3j''-  ^--^—i-     3_i      -—2— 
-=12 ;  >  531441-1_531440^     ^^,^ 

;     ~      2      ~    2 

Ex.  2.  Find  three  geometric  means  between  2  and  32. 
Here  a=zi 


wi=3  ;  ) 


and  the  means  required  are  4,  8,  16. 
Ex.  3.  The  first  term  of  a  geometrical  progression  is  1, 
the  ratio  2,  and  the  number  of  terms  10.     What  is  the  sum  of 
the  series?  •  Ans.  1023 


PROPORTION  AND  PROGRESSION.        307 

Ex.  4.  In  a  geometrical  progression  is  given  the  greatest 
term  =1458,  the  ratio  =3,  and  the  number  of  terras  =7,  to 
find  the  least  term.  Aus.  2: 

Ex.  5.  It  is  required  to  find  two  geometrical  proportionals 
between  3  and  24,  and  four  geometrical  means  between  3  and 
96.  Ans.  6  and  12  ;  and  6,  12,  24,  and  48. 

Ex.  6.  Find  two  geometric  means  between  4  and  256. 

Ans.  16,  and  64. 

Ex.  7.  Find  three  geometric  means  between  ^  and  9. 

Ans.  J,  1,  3. 

Ex.  8.  A  gentleman  who  had  a  daughter  married  on  New- 
year's  day,  gave  the  husband  towards  her  portion  4  dollars, 
promising  to  triple  that  sum  the  first  day  o.  every  month,  for 
nine  months  after  the  marriage  ;  the  sum  paid  on  the  first  day 
of  the  ninth  month  was  26244  dollars.  What  was  the  lady's 
fortune  ?  Ans.  39364  dollars. 

Ex.  9.  Find  the  value  of  l+i4-7-f-J+  &C.  ad  infinitum. 

Ans.  2. 

Ex.  10.  Find  the  value  of  H-J+t\+It4-  <^c.  ad  infini- 

Ans.  4. 


§  III.     HARMONICAL  PROPORTION  AND  PROGRESSION. 

435.  Three  quantities  are  said  to  be  in  harmonical  propor- 
tion, when  the  first  is  to  the  third,  as  the  difference  between 
the  first  and  second  is  to  the  difierence  between  the  second 
and  third. 

Thus,  a,  h,  c,  are  harmonically  proportional,  when 
a  :  c  :-.  a—h  :  h — c,  or  a  :  c  :  :  h — a  :  c  —  h. 
And  c,  [since  a(6— c)=c(a— 6)  or  ah={2a—h)c\,  is  a  third 

harmonical  proportion  to  a  and  h,  when  c=z -. 

436.  Four  quantities  are  in  harmonical  proportion,  when  the 
first  is  to  the  fourth,  as  the  difference  between  the  first  and 
second  is  to  the  difference  between  the  third  and  fourth. 

Thus,  a,  6,  c,  df,  are  in  harmonical  proportion,  when 

a  :  d  ::  a—b  :  c—d,  or  a  :  d  :  :  b  —  a  :  d—c. 
And  d,   [since  a{c—d)  =  d(a'—b)    or  ac={2a—b)d],   is  a 

fourth  harmonical  proportional  to  a,  b,  c,  when  d=- 7. 

In  each  of  which  cases,  it  is  obvious,  that  twice  the  first 
term  must  be  greater  than  the  second,  or  otherwise  the  pro- 
portionality will  not  subsist. 


308        PROPORTION  AND  PROGRESSION. 

437.  Any  number  of  quantities,  c,  h,  c,  d^  e,  «fcc.  are  in  har- 
monical  progression,  if  a  :  c  : :  a — b  :  6— c;  b  :  d  :  :  b—c  : 
c — d  ;  c  :  e  : :  c— (f  :  d — e,  &c. 

438.  The  reciprocal  of  quantities  in  harmonical  progression, 
are  in  arithmetical  progression.  For,  if  a,  A,  c,  d,  e,  &c.  are 
in  harmonical  progression  ;  then,  from  the  preceding  Article, 
we  shall  have  ^c-\-ab^=i2ac  ;  dc-\-bc=2db  ;  ed-[-cd—2ec, 
&c.     Now,  by  dividing  the  first  of  these  equalities  by  abc  ; 

the  second  by  bdc:  the  third  by  cde  ;  &c.,  we  have,  — 1--  = 

a     c 

2112112.  ^.        .        11111- 

T ;  t+:j=-  ;  -+-=:7 ;  <^c.       Therefore,  -,  -,  -,  -,  -,  &c. 
o     b     a      c     c     e     d  abode 

are  in  arithmetical,  progression. 

439.  An  harmonical  mean  between  any  two  quantities^  is  equal 
to  twice  their  product  divided  by  their  sum.  For,  if  a,  x,  b, 
are  three  quantities  in  harmonical  proportion,  then,  a  :  h  : : 

a — X  :  X — b  ;    .'.ax  —  abz=ab  —  bx,  and  x= r. 

a-\-o 

Ex.  1.  Find  a  third  harmonical  proportional  to  6  and  4. 

Let  x=  the  required  number,  then  6  :  x  ::  6—4  :  4 — x; 

.-.  2^  —  6x=2x,  and  x=2. 

Ex.  2.  Find  an  harmonical  mean  between  12  and  6. 

Ans.  8. 

Ex.  3.  Find  a  third  harmonical  proportional  to  234  and 
144.  Ans.  104. 

Ex.  4.  Find  a  fourth  harmonical  proportional  to  16,  8,  and 
3.  Ans.  2. 


^  IV.    PROBLEMS  IN  PROPORTION  AND  PROGRESSION. 


Prob.  1.  There  are  two  numbers  whose  product  is  24,  and 
the  difference  of  their  cubes  :  cube  of  their  difference  :  :  19  :  1. 
What  are  the  numbers  ' 


Let  x=  the  greater  number,  and  y=  the  lesser. 

Then,  a;y=24,  and  x^—y^  :  {x—yY  : :  19 

By  expansion,  x^ — y^  :  x^ — 3x'^y-^3xy^—y^  : :  19 

.-.  3x^y—3xy^  :  (x—yY  :  :  18 

and,  dividing  by  x^y,  3xy  :  {x—y)'^  : :  18 

but  xy=24:  i  .:  72  :  {x-y)^  : :  18 


PROPORTION  AND  PROGRESSION.  309 

Hence,  18  (x-yf=12,  or  (a;— y)2=4  ; 

.•.  X — y=2. 

Again,  x'^^2xy-\-y'^=i  4, 

and         Axy         =96, 

.•.aj2+2ry+y2=i00,  and  a:-fy=10, 
but  x—y=:  2, 


.•.a:=6,  and  y=4. 

Prob.  2.  Before  noon,  a  clock  which  is  too  fast,  and  points 
to  afternoon  time,  is  put  back  five  hours  and  forty  minutes  ; 
and  it  is  observed  that  the  time  before  shown  is  to  the  true  time 
as  29  to  105.     Required  the  true  lime. 

Ans.  8  hours,  45  ra-inutes 

Prob.  3.  Find  two  numbers,  the  greater  of  which  shall  be 
to  the  less  as  their  sum  to  42,  and  as  their  difference  to  6. 

Ans.  32,  and  24. 

Prob.  4.  What  two  numbers  are  those,  whose  difference, 
sum,  and  product,  are  as  the  numbers  2,  3,  and  5,  respectively  ? 

Ans.  10,  and  2. 

Prob.  5.  In  a  court  there  are  two  square  grass-plots  ;  a  side 
of  one  of  which  is  10  yards  longer  than  the  other ;  and  their 
areas  are  as  25  to  9.    What  are  the  lengths  of  the  sides  ? 

Ans.  25,  and  15  yards. 
Prob.  6.  There  are  three  numbers  in  arithmetical  progres- 
sion, whose  sum  is  21  ;  and  the  sum  of  the  first  and  second 
is  to  the  sum  of  the  second  and  third  as  3  to  4.    Required  the 
numbers. 

Ans.  5,  7,  9. 

Prob.  7.  The  arithmetical  mean  of  two  numbers  exceeds 
the  geometrical  mean  by  13,  and  the  geometrical  mean  ex- 
ceeds the  harmonical  mean  by  12.    What  are  the  numbers  ? 

Ans.  234,  and  104. 

Prob.  8.  Given  the  sum  of  three  numbers,  in  harmonical 
proportion,  equal  to  26,  and  their  continual  product  =576  ;  to 
find  the  numbers. 

Ans.  12,  8  and  6. 

Prob.  9.  It  is  required  to  find  six  numbers  in  geometrical 
progression,  such,  that  their  sum  shall  be  315,  and  the  sum  of 
the  two  extremes  165. 

Ans.  5,  10,  20,  40,  80,  and  160. 

Prob.  10.  A  number  consisting  of  three  digits  which  are  in 


310  PROPORTION  AND  PROGRESSION. 

arithmetical  progression,  being  divided  by  the  sum  of  its  di- 
gits, gives  a  quotient  48  ;  and  if  198  be  subtracted  from  it,  the 
digits  will  be  inverted.     Required  the  number. 

Ans.  432. 
Prob.  11.  The  difference  between  the  first  and  second  of 
four  numbers  in  geometrical  progression  is  36,  and  the  diffe- 
rence between  the  third  and  fourth  is  4  ;  What  are  the  num- 
bers ? 

Ans.  54,  18,  6,  and  2. 
Prob.   12.  There  are  three   numbers  in  geometrical  pro- 
gression ;  the  sum  of  the  first  and  second  of  which  is  9,  and 
the  sum  of  the  first  and  third  is  15.    Required  the  numbers. 

Ans.  3,  6,  12. 
Prob.  13.  There  are  three  numbers  in  geometrical  pro- 
gression, whose  continued  product  is  64,  and  the  sum  of  their 
cubes  is  584.     What  are  the  numbers  1 

Ans.  2,  4,  8. 
Prob.  14.  There  are  four  numbers  in  geometrical  progres- 
sion, the  second  of  which  is  less  than  the  fourth  by  24  ;    and 
the  sum  of  the  extremes  is  to  the  sum  of  the  means  as  7  to  3. 
Required  the  numbers. 

Ans  1,  3,  9,  27. 
Prob.  15.  There  are  four  numbers  in  arithmetical  progres- 
sion, whose  sum  is  28  ;  and  their  continued  product  is  585. 
Required  the  numbers. 

Ans.  1,  5,  9,  13. 
Prob.   16.  There  are  four  numbers  in  arithmetical  progres- 
sion ;  the  sum  of  the  squares  of  the  first  and  second  is  34  ; 
and  the  sum  of  the  squares  of  the  third  and  fourth  is  130. 
Required  the  numbers. 

Ans.  3,  5,  7,  9. 


CHAPTER  XIV. 


ON    LOGARITHMS. 

440.  Previous  to  the  investigation  of  Logarithms,  it  may 
not  be  improper  to  premise  the  two  following  propositions. 

441.  Any  quantity  which  from  positive  becomes  negative,  and 
reciprocally,  passes  through  zero,  or  infinity.  In  fact,  in  order 
that  m,  which  is  supposed  to  be  the  greater  of  the  two  quantities 
m  and  n,  becomes  n,  it  must  pass  through  n  ;  that  is  to  say, 
the  difference  m  —  n  becomes  nothing  ;  therefore  p,  being  this 
difference,  must  necessarily  pass  through  zero,  in  order  to 
become  negative,  or  —p.  But  if  p  becomes  —p,  the  fraction 
jf  will  become  —  ^  ;  and  therefore  it  passes  through  ^,  or  in- 
finity. 

442.  It  may  be  observed,  that  in  Logarithms,  and  in  some 
trigonometrical  lines,  the  passage  from  positive  to  negative  is 
made  through  zero  ;  for  others  of  these  lines,  the  transition" 
takes  place  through  infinity  :  It  is  only  in  the  first  case  that 
we  may  regard  negative  numbers  as  less  than  zero  ;  whence 
there  results,  that  the  greater  any  number  or  quantity  a  is, 
when  taken  positively,  the  less  is  —a ;  and  also,  that  any  ne- 
gative number  is,  a  fortiori,  less  than  any  absolute  or  positive 
number  whatever. 

443.  If  we  add  successively  different  negative  quantities  to 
the  same  positive  magnitude,  the  results  shall  be  so  much  less 
according  as  the  negative  quantity  becomes  greater,  abstract- 
ing from  its  sign.     For  instance,  8  — 1>8— 2>8— 3,  &c. 

It  is  in  this  sense,  that  0>  —1  >  — 2>  ~3,  &c. ;  and  3> 
0>-l>-2>-3>-4,  &c. 

444.  Any  quantity,  which  from  real  becomes  imaginary,  or 
reciprocally ,  passes  through  zero,  or  infinity.  This  is  what  may 
easily  be  concluded  from  these  expressions, 

considered  in  these  three  relations, 


312  ON  LOGARITHMS. 


^  I.    THEORY  OF  LOGARITHMS. 

445.  Logarithms  are  a  set  of  numbers,  which  have  been 
compnted  and  formed  into  tables,  for  the  purpose  of  facilitat- 
ing arithmetical  calculations  ;  being  so  contrived,  that  the  ad- 
dition and  subtraction  of  them  answer  to  the  multiplicatioa 
and  division  of  the  natural  numbers,  with  which  they  are  made 
to  correspond. 

446.  Or,  when  taken  in  a  similar,  but  more  general  sense, 
logarithms  may  be  considered  as  the  exponents  of  the  pow- 
ers, to  which  a  given,  or  invariable  number,  must  be  raised, 
in  order  to  produce  all  the  common,  or  natural  numbers. 
Thus,  if  af—y^  a"z=zy\  a"" =y^' ,  &c. ;  then  will  the  indices 
a:,  X',  x'\  &c.  of  the  several  powers  of  a,  be  the  logarithms  of 
the  numbers  y,  y,  y",  &c.  in  the  scale  or  system,  of  which  a 
is  the  base. 

447.  So  that,  from  either  of  these  formulae,  it  appears,  that 
the  logarithm  of  any  num.ber,  taken  separately,  is  the  index 
of  that  power  of  some  other  number,  which,  when  it  is  involved 
in  the  usual  way,  is  equal  to  the  given  number.  And  since 
the  base  a,  in  the  above  expressions,  can  be  assumed  of  any 
value,  greater  or  less  than  1,  it  is  plain  that  there  may  be  an 
endless  variety  of  systems  of  logarithms,  answering  to  the 
same  natural  numbers. 

448.  Let  us  suppose,  in  the  equation  a'z^y,  at  first,  a:=0, 
we  shall  havey=l,  since  a^z^l  ;  to  a?=z:l,  corresponds  yz=a. 
Therefore,  in  every  system^  the  logarithm  of  unity  is  zero  ;  and 
also,  the  base  is  the  number  whose  proper  logarithm^  in  the  sys- 
tem to  which  it  belongs,  is  unity.  These  properties  belong  es- 
sentially to  all  systems  of  logarithms. 

449.  Let  -{-X  he  changed  into  — x  in  the  above  equation, 
and  we  shall  have 

1 

Now,  the  exponent  x  augmenting  continually,  the  fraction 

— ,  if  the  base  a  be  greater  than  unity,  will  diminish,  and  may 

be  made  to  approach  continually  towards  0,  as  its  limit ;  to 
this  limit  corresponds  a  value  of  x  greater  than  any  assignable 
number  whatever.  Hence  it  follows,  that,  when  the  base  a  is 
greater  than  unity,  the  logarithm  of  zero  is  infinitely  negative. 

450.  Let  y  and  y'  be  the  representatives  of  two  numbers, 
X  and  x'  the  corresponding  logarithms  for  the  same  base  :   we 


ON  LOGARITHMS.  313 

shall  have  these  two  equations,  a'—y,  and  a"—y',  whose  pro- 
duct is  a'-a^'^y-y',  or  a"^^'  =yy\  and  consequently,  by  the  de- 
finition of  logarithms,  a;+,r'=:log.  yy\  or  log.  yy'  =  \og.  y-|- 
log.  y\ 

And,  for  a  like  reason,  if  any  number  of  the  equations 
(f=y,  af'—y',  a'"—y'^^  &c.  be  multiplied  together,  we  shall 
have  a:-^+^^+^^''+^"'- =yy'y',  &;c.  ;  and,  consequently,  a;+aj''4-^'^ 
&LC.  =  log.  yy'y'\  &c.  ;  or  log.  yy'y'\  &c.=:log.  y+log-y'+ 
log.  y'\  &c. 

The  logarithm  of  the  product  of  any  number  of  factors  isj 
therefore,  equal  to  the  sum  of  the  logarithms  of  those  factors. 

451.  Hence,  if  all  the  factors  y,  y\  y'\&Lc.  are  equal  to 
each  other,  and  the  number  of  them  be  denoted  by  m,  the  pre- 
ceding property  will  then  become  log.  {y"')  =  m,  log.  y. 

Therefore,  the  logarithm  of  the  mth  power  of  any  number  is 
equal  to  m  times  the  logarithm  of  that  number. 

452.  In  like  manner,  if  the  equation  a'=y,  be  divided  by 

a" 
<f'—y\   we    shall  have,  from  the  nature  of  powers,  —  ,   or 

a       =— ,  ;  and  by  the  definition  of  logarithms,  a;  — a;'=log 

y 

(|r)  5  0^  log.y-log.  y'=log.  {y^y 

Hence  the  logarithm  of  a  fraction,  or  of  the  quotient  arising 
from  dividing  one  number  by  another,  is  equal  to  the  logarithm 
of  the  numerator  minus  the  logarithm  of  the  denominator. 

453.  And  if  each  member  of  the  equation,  a'~y,  be  rais- 

nix  m 

ed  to  the  fractional  power  ^,  we  shall  have  a"  —y"  ;  and 
consequently,  as  before,  — a:r=log.  (y")~log.  y  y""  ;    or,  log. 


!l     m  , 
r=  — log.  y. 

Therefore,  the  logarithm  of  a  mixed  root,  or  power,  of  any 
number,  is  found  by  multiplying  the  logarithm  of  the  given 
number,  by  the  numerator  of  the  index  of  that  power,  and  divi- 
ding the  result  by  the  denominator. 

454.  And  if  the  numerator  m  of  the  fractional  index  of  the 
number  y,  be,  in  this  case,  taken  equal  to  1,  the  preceding 
formula  will  then  become 

log.  y"=:i  log.  y. 
From  which  it  follows,  that  the  logarithm  of  the  nth  root  of 
28 


314  ON  LOGARITHMS. 

any  number,  is  equal  to  the  nth  part  of  the  logarithm  of  that 
number. 

455.  Hence,  besides  the  use  of  logarithms  in  abridging  the 
operations  of  multiplication  and  division,  they  are  equally  ap- 
plicable to  the  raising  of  powers  and  extracting  of  roots  ; 
which  are  performed  by  simply  multiplying  the  given  loga- 
rithm by  the  index  of  the  power,  or  dividing  it  by  the  number 
denoting  the  root. 

456.  But,  although  the  properties  here  mentioned  are  com- 
mon to  every  system  of  logarithms,  it  was  necessary  for 
practical  purposes  to  select  some  one  of  these  systems  from 
the  rest,  and  to  adapt  the  logarithms  of  all  the  natural  num- 
bers to  that  particular  scale.  And  as  10  is  the  base  of  our 
present  system  of  arithmetic,  the  same  number  has  accord- 
ingly been  chosen  for  the  base  of  the  logarithmic  system  now 
generally  used. 

457.  So  'chat,  according  to  this  scale,  which  is  that  of  the 
common  logarithmic  tables,  the  numbers, 

—4—3—2—10  1  2  3  4 

etc.  10     ,10     ,10     ,10     ,  10  ,    10  ,     10  ,     10  ,     10  , 
etc.  ;  or, 

""=•  10500'  li-  rSo-   1^' ''  '"'  ""'•  ^''°<''  1°°"°' 

etc.,  have  for  their  logarithms, 
etc.  —4,  —3,  —2,  —1,  0,   1,  2,  3,  4,  etc, 
which  are  evidently  a  set  of  numbers  in  arithmetical  progres- 
sion, answering  to  another  set  in  geometrical  progression  ;  as 
is  the  case  in  every  system  of  logarithms. 

458.  And,  therefore,  since  the  common  or  tabular  logarithm 
of  any  number  (n)  is  the  index  of  that  power  of  10,  which, 
when  involved,  is  equal  to  the  given  number,  it  is  plain,  from 
the  equation  W=n,  or  10-^=-!,  that  the  logarithms  of  all  the 
intermediate  numbers,  in  the  above  series,  may  be  assigned 
by  approximation,  and  made  to  occupy  their  proper  places  in 
the  general  scale. 

459.  It  is  also  evident  that  the  logarithms  of  1,  10,  100, 
1000,  etc.,  being  0,  1,2,  3,  respectively,  the  logarithm  of  any 
number,  falling  between  1  and  10,  will  be  0,  and  some  deci- 
mal parts  ;  that  of  a  number  between  10  and  100,  1  and  some 
decimal  parts  ;  of  a  number  between  100  and  1000,  2  and  some 
decimal  parts  ;  and  so  on. 

460.  And,  for  a  like  reason,  the  logarithms  of  — ,    r-^, 


ON  LOGARITHMS.  315 

— -,  etc.  or  of  their  equals,  .1,  .01,  .001,  etc.  in  the  de- 
scending part  of  the  scale,  being  —1,  —2,  —3,  etc.  the  loga- 
rithm of  any  number,  falling  between  0  and  .1,  will  be  —1  and 
some  positive  decimal  parts  ;  that  of  a  number  between  .1  and 
.01^  —2  and  some  positive  decimal  parts  ;  and  so  on. 

461.  Hence,  as  ihe  multiplying  or  dividing  of  any  number 
by  10,  100,  1000,  etc.  is  performed  by  barely  increasing  or 
diminishing  the  integral  part  of  its  logarithm  by  1,2,  3,  &c. 
it  is  obvious  that  all  numbers  which  consist  of  the  same 
figures,  whether  they  be  integral,  fractional,  or  mixed,  will 
have  the  same  quantity  for  the  decimal  part  of  their  loga- 
rithms. Thus,  for  instance,  if  i  be  made  to  denote  the  index, 
or  integral  part  of  the  logarithm  of  any  number  N,  and  d  its 
decimal  part,  wc  shall  have  log.  N=24-c? ;  log.  10"'XN  = 

N  * 

(i-f  m)4-cZ ;  log.  —^==^(i — 'm)-\-d  ;  where  it  is  plain  that  the 

decimal  part  of  the  logarithm,  in  each  of  these  cases,  remains 
the  same. 

462.  So  that  in  this  system,  tne  integral  part  of  any  loga- 
rithm, which  is  usually  called  its  index,  or  characteristic,  is 
always  less  by  1  than  the  number  of  integers  which  the  natu- 
ral number  consists  of ;  and  for  decimals,  it  is  the  number 
which  denotes  the  distance  of  the  first  significant  figure  from 
the  place  of  units.  Thus,  according  to  the  logarithmic  tables 
in  common  use,  we  have 


lumbers. 
1.36820 

Logarithms 
0.1361496 

335.260 

2.5253817 

.46521 

1.6676490 

.06154 

2.7891575 

&c. 

'&c. 

where  the  sign  —  is  put  over  the  index,  instead  of  before  it, 
when  that  part  of  the  logarithm  is  negative,  in  order  to  distin- 
guish it  from  the  decimal  part,  which  is  always  to  be  consi- 
dered as  +,  or  affirmative. 

463.  Also,  agreeably  to  what  has  been  before  observed,  the 
logarithm  of  38540  being  4.5859117,  the  logarithms  of  any 
other  numbers,  consisting  of  the  same  figures,  will  be  as  fol- 
lows • 


316  ON  LOGARITHMS. 

Numbers.  Logarithms 

3854  3.5859117 

385.4  2.5859117 

38.54  1.5859117 

3.854  0.5859U7 

.3854  L5859117 

.03854  2.5859117 

.003854  3.5859117 

which  logarithms,  in  this  case,  differ  only  in  their  indices,  the 
decimal  or  positive  part,  being  the  same  in  them  all. 

464.  And  as  the  indices,  or  the  integral  parts  of  the  loga- 
rithms of  any  numbers  whatever,  in  this  system,  can  always 
be  thus  readily  found,  from  the  simple  consideration  of  the 
rule  above-mentioned,  they  are  generally  omitted  in  the  ta- 
bles, being  left  to  be  supplied  by  the  operator,  as  occasion  re- 
quires. 

465.  It  may  here,  also,  be  farther  added,  that,  when  the 
logarithm  of  a  given  number,  in  any  particular  system,  is 
known,  it  will  be  easy  to  find  the  logarithm  of  the  same  num- 
ber in  any  other  system,  by  means  of  the  equations,  a'-=.n, 
e"'z=znj  which  give 

(1)     .     .     .     .     x=  log.  n,  x'=  1.  71 (2). 

Where  log.  denotes  the  logarithm  of  n,  in  the  system  of  which 
a  is  the  base,  and  1.  its  logarithm  in  the  system  of  which  e  is 
the  base. 

466.  Whence  a'=e",  or  a''=e,  and  c  z=a,  we  shall  have, 

"^  X  x^ 

for  the  base  a,  —f-=  log.  e,  and  for  the  base  c,  — =zl.a  ;  or 

(3)     .     .     .     .     x=x'  log.  c,  x'  =  x.l.a (4). 

Whence,  if  the  values  of  x  and  x',  in  equations  (1),  (2), 
be  substituted  for  x  and  x'  in  equations  (3),  (4),  we  shall  have, 

log.  n=  log.  exl.n,  and  l.n= x  log.  n  ;  or   l.n=l.a  X 

log.  e 

log.  n,  and  log.  n=j—xl.n.  where  log.  e,  or  its  equal  j-  ex- 
presses the  constant  ratio  which  the  logarithms  of  n  have  to 
each  other  in  the  systems  to  which  they  belong. 

467.  But  the  only  system  of  these  numbers,  deserving  of 
notice,  except  that  above  described,  is  the  one  that  furnishes 
what  have  been  usually  called  hyperbolic  or  Neperian  loga- 
rithms, the  base  of  which  is  2.718281828459  .  .      . 


ON  LOGARITHMS.  317 

468.  Hence,  in  comparing  this  with  the  common  or  tabular 
logarithms,  we  shall  have,  by  putting  a  in  the  latter  of  the 
above  formulae  =10,  the  expression 

log.  71—  xl.Tiy  or  Z.n  =  ?.10xlog.  n 

(.  L  U 

Where  log.,  in  this  case,  denotes  the  common  logarithm  of 
the  number  n,  and  /.  its  Neperian  logarithm ;  the  constant 

factor -^i^  which  is  ^^2-±33^g,  or  -4342944819  .  .  be- 

ing  what  is  usually  called  the  modulus  of  the  common  or  ta- 
bular system  of  logarithms. 

469.  It  may  not  be  improper  to  observe,  that  the  logarithms 
of  negative  ^antities,  are  imaginary ;  as  has  been  clearly 
proved,  by  Lacroix,  after  the  manner  of  Euler,  in  his  Traite 
du  Calcul  Differentiel  et  Integral ;  and  also,  by  Suremain-Mis- 
SERV  in  his  Theorie  Purement  Algebrique  des  Quantites  Ima- 
ginaires.  See,  for  farther  details  upon  the  properties  and  cal- 
culation of  logarithms,  Garnier's  dAlgehre,  or  Bonnycastle's 
Treatise  on  Algebra  in  two  vols.  8vo. 


^  II.  APPLICATION  OF  LOGARITHMS  TO  THE  SOLUTION  OF  EXPO- 
NENTIAL EQUATIONS. 

470.  Exponential  equations  are  such  as  contain  quanti- 
ties with  unknown  or  variable  indices  :    Thus,  a^-^b,  o(^z=Cf 

V 

— =c?,  &c.  are  exponential  equations 

471.  An  equation  involving  quantities  of  the  form  o?^,  where 
the  root  and  the  index  are  both  variable,  or  unknown,  seldom 
occur  in  practice,  we  shall  only  point  out  the  method  of  solv- 
ing equations  involving  quantities  of  the  form  a^,  a*^,  where 
the  base  a  is  constant  or  invariable. 

472.  It  is   proper  to  observe  that  an  exponential  of  the 

form  a   ,  means,  a  to  the  power  of  5*,  and  not  ab  to  the  power 
ofx. 

Ex.  1.  Find  the  value  of  x  in  the  equation  a'^^b. 

Taking  the   logarithm  of  the  equation  a^^=^b^  we  have  a:X 

log.  a=log.  b  ;  .-.0:=,—^  ;    thus,  let  cr  =  5,  J  =  100  ;  then  in 
log.  a 

the  equation  5^=100, 

—  ^og-  100     2.0000000  _ 

*""     log.  5~'~0.698976o"~  * 

28* 


318  ON  LOGARITHMS 

Ex.  2.  It  is  required  to  Jind  the  value  of  x  in  the  equation 

Assume  bx=y,  then  ay=Cj  and  yxlog.  a=\og.  c;  .*.  y= 

~^.     Hence  b^=^^  (which  \ei)—d.     Take    the  loga- 
log.  a  log.  a  ^  '  ^ 

rithm  of  the    equation   b'=d,    then,  by  (Ex.  1),  x=^^^. 

Thus,  let  a  =  9,  b=3,  0=1000  ;  then  in  the  equation  9^= 

,o^^   log.  c     log.  1000     „  ,  ,,      „  ,        log.  d 

1000,  _§_      _^—  =3.14  =(^   ;  and  a=r-^-.= 
log.  &        log.  9  \       /  '  jQg^  ^ 

log.  3' 14     .4969296       ,  ^^ 

z^: —  1  04 

log.  3        .4771213  "~   ■ 
Ex.  3.  Make  such  a  separation  of  the  quantities  in  the  equa- 
tion (a'^—b^Yz=a-\-b,  as  to  show,  that  _^_  ^g-  ;^"^  J. 

l—x     log.  (a  —  b) 
Taking  the  logarithm,  we  have 
xxlog.  (a'^  —  b^)  =  log.  (a-{-b),  or  xxlog.  (c+6)  X  {a—b)  = 
log.  (a-{-b)  ; 
that  is,  a;Xlog.  (a4-fe)+a:Xlog.  (a  — 5)=log.  (a-irb). 

Hence   a^xlog.   (a  —  b)—\og.  {a -^  b)  —xxlog.  {a-{- b)  = 

0-.) .0,  (.+.).  ...^^=5ig|). 

Ex.  4.  Given  a'-\-b^=Cj  and  af—b^=d,  required  the  va- 
lues of  X  and  y. 

By  addition,  2a^=c-\-d,  or  a'=— - — ,  which  put  =m  ;  then 

log.  m 

"fog.  a  ' 

c—d 
Again,  by  subtraction,   we    have   2fty=c— c?,    or    b^=. 


(which  let  =n)  ;  .*.  y: 


2 

log'  y» 
log.  b' 

ab^-\-c 


Ex.  5.  Find  the  value  of  x  in  the  equation 

Ans.  x=: 


d 
log.  (de—c)—\og.  a 


log.  6 

Ex.  6.  Find  the  value  of  x  in  the  equation  a'= — —-5 . 

Ans  ■,- JlQg-  (^+^)+i  log-  (^-^)--7  log-  ^— I  l»g'  <^ 

log.  a 


ON  LOGARITHMS.  319 

Ex.  7.  Find  the  value  of  x  in  the  equation  i(^-\--^=YsCi^ 

1 

4-1.  Ans.  x=. . 

^  log.  a 

Ex.8.  Given  log.  a?  4- log- y=i  Mo  ^^^   the   values  of   x 
and  log.  aj—log.  y  =  iS  and  y. 

Ans.  a:=:10  ^10,  and  y=:10. 
Ex.  9.  In  the  equation  2'=:  10,  it  is  required  to  find  the  va- 
lue of  X.  Ans.  a-rrrS. 321928,  <fec. 
Ex.  10.  Given  y  729=3,  to  find  the  value  of  x. 

Ans.  x=6. 
Ex.  11.  Given  ^  57862  =  8,  to  find  the  value  of  x. 

Ans.  a?=5.2735,  &c 

3 

Ex.  12.    Given  (216)'  =64,  to  find  the  value  of  x. 

Ans.  a;= 3.8774,  &c 

Ex.  13.    Given  43=4096,  to  find  the  value  of  x. 

Ans.  a:=r^^  =  1.6309,  &c. 
log.  3 

Ex.  14.  Given  a'^^=c,  3.ndb'-''=d,  to  find  the  values  of 

X  and  y. 

.  m-\-n        ,         m—n        .  log.  c 

Ans.  =— — — ,  and  y=———  :  where  m=T-^ — ,   and  n= 
2    '         ^         2      '  log.  a' 

log<^ 
log.  b' 


CHAPTER   XV. 

ON 

THE    RESOLUTION    OF    EQUATIONS 

OF  THE  THIRD  AND  HIGHER  DEGREES. 

§  I.  THEORY  AND  TRANSFORMATION  OF  EQUATIONS. 

473.  In  addition  to  what  has  been  already  said  (Art.  168), 
it  may  here  be  observed,  that  the  roots  of  any  equation  are 
the  numbers,  which,  when  substituted  for  the  unknown  quan- 
tity, will  make  both  sides  of  the  equation  identically  equal.  Or, 
which  is  the  same,  the  roots  of  any  equation  are  the  numbers, 
which,  substituted  for  the  unknown  quantity,  reduce  the  first 
member  to  zero,  or  the  proposed  equation  to  the  form  of  0  =  0  ; 
because  every  equation  may,  designating  the  highest  power  of 
the  unknown  quantity  by  a:"*,  be  exhibited  under  the  form 

a:'"+Aa;'»-'4-Ba;'"-2+Ca:'"-3+   .   .   .  Ta;4-V  =  0.  (1), 
A,B,  C,  ...  T,  V,  being  known  quantities.     And  the  resolu- 
tion of  an  equation  is  the  method  of  finding  all  the  roots,  which 
will  answer  the  required  condition. 

474.  This  being  premised,  it  may  now  be  shown,  that  if  a 
he  a  root  of  the  equation  (1),  the  left-hand  member  of  that  equo' 
tion  will  he  exactly  divisible  by  x  —  a. 

For  if  a  be  substituted  for  x,  agreeably  to  the  above  defini- 
nition,  we  shall  necessarily  have 

flr  +  Aa'"-i  +  Ba'--2+Ca'»-34-   .  .  .  Ta+V  =  0. 
And  consequently,  by  transposition, 

V^-a*"— Aa^-i-Ba''-^— Ca--^—   .  .  .  —To. 
Whence,  if  this  expression  be  substituted  for  V  in  the  first 
^  equation,. we  shall  have,  by  uniting  the  corresponding  terms, 
and  placing  them  all  in  a  line, 

(a?-— a'")4.A(a;'"-^-a'"-^)-f-B(a:"-2-a'»-3)4-T(x-a)  =  0. 
Where,  since  the  difference  of  any  two  equal  powers  of 
two  different  quantities  is  divisible  by  the  difference  of  their 


RESOLUTION  OF  EQUATIONS.  321 

roots  (Art.  108),  each  of  the  quantities  (x"'—a'"),  (x^-  — a"*— i), 
(x'^—2  —  a^—^),  &;c.  will  be  divisible  by  x—a.  And,  therefore, 
the  whole  compound  expression 

(a;"'_a'")-f-A(.x'«-i  — a'»-^)  +  B(a;'«-2— a'»-2)+  &c.  =0, 
which  is  equivalent  to  the  equation  first  proposed,  is  also  di- 
visible  by  x—a  ;  as  was  to  be  shown. 

But  if  a  be  a  quantity  greater  or  less  than  the  root,  this 
conclusion  will  not  take  place  ;  because,  in  that  case,  we  shall 
not  have 

V=— a'"— Aa'»-i  — Ba"«-2— Ca'"-^—  ....  —Ta; 
which  is  an  equality  obviously  essential  to  the  division  in 
question. 

475.  The  preceding  proposition  may  be  demonstrated,  af- 
ter the  manner  of  D'Alembert,  as  follows  :  In  fact,  desig- 
nating by  X,  the  polynomial,  which  forms  the  first  member  of 
the  equation  (1) ;  then  we  shall  always  carry  on  the  division 
of  X  by  x  —  a,  till  we  arrive  at  a  remainder  R,  independent  of 
X,  since  x  is  only  of  the  first  degree  in  the  divisor ;  so  that, 
representing  by  Q  the  corresponding  quotient,  we  shall  have 
this  identity, 

X  =  Q(a;-a)  +  R. 

Now,  by  hypothesis,  a  substituted  for  x  reduces  the  poly- 
nomial X  to  zero  ;  and  it  is  evident  that  the  same  substitution 
gives  Q(x—a)  =  0  ;  therefore  we  shall  necessarily  have  0  =  R  : 
Hence  x—a  divides  the  equation  (1),  without  a  remainder. 

Reciprocally,  if  the  first  member  of  any  equation  of  the  form 
X=zO  be  divisible  by  x  —  a,  a,  is  a  root.  In  fact  we  have,  accord- 
ing to  this  hypothesis,  the  identity  X=:Q{x—a),  which,  for 
x=a,  gives  X  =  0  ;  therefore,  (Art.  473),  a  is  a  root  of  the 
proposed  equation. 

CoR.  1.  Hence  we  may  easily  conclude,  that  if  a  |?e  not  a 
root  of  the  equation  (1),  the  first  member  will  not  be  divisible 
by  x—a. 

Cor.  2.  And  if  the  first  member  of  the  equation  (1),  be 
not  divisible  by  x—a,  a  is  not  a  root  of  the  proposed  equa- 
tion. 

476.  Supposing  every  equation  to  have  one  root,  or  value  of 
the  unknown  quantity,  it  can  then  be  shown,  that  any  proposed 
equation  will  have  as  many  roots  as  there  are  units  in  the  index 
of  its  highest  term,  and  no  more.  For  let  a,  according  to  the 
assumption  here  mentioned,  be  a  root  of  the  equation  (1), 

x^-\  Aa;"»-i4-Ba;  ~'-+Cx'^-^-i-  .  .  .  +Ta;-1-V=:=0. 


322  RESOLUTION  OF  EQUATIONS. 

Then,  since  by  the  last  proposition  this  is  divisible  by  x—Oj 
it  will  necessarily  be  reduced,  by  actually  performing  the  ope- 
ration, to  an  equation  of  the  next  inferior  degree,  or  one  of  the 
former 

And  as  this  equation,  by  the  same  hypothesis,  has  also  a  root, 
which  may  be  represented  by  a',  it  will  likewise  be  reduced, 
when  divided  by  x — a',  to  another  equation  one  degree  lower 
than  the  last ;  and  so  on. 

Whence,  as  this  process  can  be  continued  regularly  in  the 
same  manner,  till  we  arrive  at  a  simple  equation,  which  has 
only  one  root,  it  follows  that  the  proposed  equation  will  have 
m  roots 

a,  a\  a'\  a''', a('"-i)' ; 

and  that  its  successive  divisors,  or  the  factors  of  which  it  is 
composed,  will  be 

X — a,  X — a',  X — a\  x — a'",  ....  x—c['^~'^)\ 
being  equal  in  number  to  the  units  contained  in  the  index  m 
of  the  highest  term  of  the  equation. 

Cor.  If  the  last  term  of  an  equation  vanishes,  as  in  the 
form  a:'"  +  Aa;'"~^4-Ba;'"~2_}_  _  _  _^Ta;=0,  it  is  evident  that 
a;=0  will  satisfy  the  proposed  equation ;  and  consequently  0 
is  one  of  its  roots.  And  if  the  two  last  terms  vanish,  or  the 
equation  be  of  the  form  x^-\- kx^~'^-{-Bx'^~'^ -\-  .  .  .  +Sa:2=:0, 
two  of  its  roots  are  0  ;  and  so  on.  See,  'for  another  demon- 
stration of  the  preceding  proposition,  Bonny  castle's  Algebra^ 
vol.  ii.  8vo. 

477.  Since  it  appears  (Art.  474),  that  every  equation, 
when  all  its  terms  are  brought  to  one  side,  is  exactly  divi- 
sible by  the  unknown  quantity  in  that  equation  minus  either  of 
its  roots,  and  by  no  other  simple  factor,  it  is  evident  that  the 
equation 

iC^+Aa:'^  i+Bx'«-2+Ca;'"-3-h  .  .  Tj:+V=:0  .  (1), 
of  which  a,  b,  c,  d,  .  .  .  Z,  are  supposed  to  be  its  several  roots, 
is  composed  of  as  many  factors 

{x-a)  {x-b)  {x-c)  {x-d)  .  .  {x-l)  .  (2), 
as  the  equation  has  roots  ;  and  that  it  can  have  no  other  factor 
whatever  of  that  form. 

478.  Whence,  as  these  two  expressions  are,  by  hypothe- 
sis, identical,  the  proposed  equation,  by  actually  multiplying 


RESOLUTION  OF  EQUATIONS. 


323 


the   above   factors,  and  arranging  the  terms  according  to  the 
powers  of  a;,  will  become 


x'"—a 

xm—\  -j-  ab 

a;m— 2_Q5jc 

x^-\abc.l)=iO 

-b 

-\-ac 

-abd 

—  c 

-\-ad. 

— acd 

-d 

-^bc 

^bcd 

&c. 

&c. 

&c. 

which  form  is  general,  whatever  maybe  the  different  signs  of 
the  roots,  or  of  the  terms  of  the  equation  ;  taking  a,  b,  c,  &c. 
as  well  as  A,  B,  C,  &;c.  in  +  or  —  as  they  may  happen  to  be. 

479.  Hence,  since  the  two  equations  (1),  (3),  are  identical, 
the  coefficients  of  the  like  powers  of  a;,  are  equal ;  and  con- 
sequently, the  following  relations  between  the  coefficients  and 
roots  will  be  sufficiently  obvious. 

I.  The  sum  of  all  the  roots  of  any  equation^  having  its  terms 
arranged  according  to  the  order  of  the  powers  of  the  unknown 
quantity,  is  equal  to  the  coe^cient  of  the  second  term  of  that 
equation,  with  its  sign  changed. 

II.  The  sum  of  the  products  of  all  the  roots,  taken  two  a^ 
two,  is  equal  to  the  coe^cient  of  the  third  term,  with  its  proper 
sign;  and  so  on. 

III.  The  continued  product  of  all  the  roots,  is  equal  to  the 
last  term,  taken  with  the  same  or  a  contrary  sign,  according  as 
the  equation  is  even  or  odd. 

480.  It  is  very  proper  to  observe,  that  we  cannot  have  all 
at  once  x—a,  x=b,  a:=c,  &;c.  for  the  roots  of  any  equation  as 
in  the  formula  (2) ;  except  when  a  =  b  —  c=d,  Slc,  ihsii  is, 
when  all  the  roots  are  equal.  The  factors  x  —  a,  x  —  b,  x—c, 
&c.  exist  in  the  same  equation  :  because  algebra  gives,  by  one 
and  the  same  formula,  not  only  the  solution  of  the  particular 
problem  from  which  that  formula  may  have  originated  ;  but 
also  the  solution  of  all  prol^ms  which  have  similar  condi- 
tions. The  different  roots  of  the  equation  satisfy  the  respect- 
ive conditions  ;  and  those  foots  may  differ  from  one  another 
by  their  quantity,  and  by  their  mode  of  existence. 

481.  To  this  we  may  likewise  add,  that,  if  the  roots  of  any 
equation  be  all  positive,  as  in  formula  (2),  where  the  factors 
are  of  the  form 

(a;_a)  [x-b)  {x-c)  (x-d)   ....  {x-l)  =  0, 
the  signs  of  the  terms  will  be  alternately  +  and  —  ;  as  will 
readily  appear  from  performing  the  operation  required. 


324  RESOLUTION  OF  EQUATIONS. 

482.  But  if  the  roots  be  all  negative,  in  which  case  the 
factors  will  be  of  the  form 

(x-\-a)  {x+b)  (x+c)  {x+d)  .  .  {x-\-l)  =  0, 
the  signs  of  all  the  terms  will  be  positive  ;  because  the  equa- 
tion arises  wholly  from  the  multiplication  of  positive  quanti- 
ties. 

Some  equations  have  their  roots  in  part  positive,  and  in  part 
negative:  Thus,  in  the  cubic  equation,  {x^a)  X  {x—b)x 
(x-{-c)  =  Oj  or  x^-\-{c—a  —  b)x^-^{ab — ac — be)  X  x-4-abc=iO, 
there  are  tviro  positive  and  one  negative  root;  because,  when 
X — a=0,  x=a  ;  x — ft=0,  x=b  ;  x-{-c=zO,  x=z—c. 

483.  Any  equation,  having  fractional  coefficients,  may  be 
transformed  into  another,  that  shall  have  the  coeffcient  of  its 
first  term  unity,  and  those  of  the  rest,  as  well  as  the  absolute 
terms,  whole  numbers. 

For  let  there  be  taken,  instead,  of  a  general  equation  of  this 
kind,  the  following  partial  example, 

which  will  be  sufficient  to  show  the  method  that  should  be 
Allowed  in  other  cases. 

Then  if  each  of  the  terms  be  multiplied  by  the  product  of 
the  denominators,  or  by  their  least  common  multiple,  we  shall 
have  12a;3  +  6a;2  4-8a;  +  9=:0,  where  the  coefficients  and  abso- 
lute term  are  all  whole  numbers. 

And  if   12a:,  in  this  case,  be^'put  =y,  or  x=y-,  there   will 

arise  by  substitution. 

Which  last  equation,  when  all  its  terms  are  multiplied  by  12^, 
gives  y3+%^4-9% 4- 1296=0  ;  where  the  coefficient  of  the 
first  term  is  unity,  and  those  ofii^he  rest  whole  numbers,  as 
was  required. 

So  that  when  the  value  of  y  in  this  equation  is  koown,  we 

V 
shall  have  for  the  proposed  equation  xz=z~-. 

1 4) 

484.  Any  equation  may  be  transformed  into  another,  the  roots 
of  which  shall  be  greater  or  less  than  those  of  the  former  by  a 
given  quantity. 

Thus,  let  there  be  taken,  as  before,  the  following  general 
equation, 


RESOLUTION  OF  EQUATIONS.  325 

.T«-f  Aa;"'-i  +  Ba?"'-24-Ca:'«-3+  .  .  Ta?4-V=0. 
And  suppose  it  were  retjuired  to  transform  it  into  another, 
whose  roots  shall  be  greater  than  those  of  the  given  equation 
by  e. 

Then,  if  y  be  made  to  represent  one  of  these  roots,  we  shall 
ha^e,  by  the  nature  of  the  question, 

y=x-{-e,  or  x=y—e. 
And,  consequently,  by  substituting  y—e  for  x,  in  the  proposed 
equation,  there  will  arise 

ym-2  ^Q _0 


y* — me 

4-A 


— (m  — l)Ae 
+B 


(4). 

which  equation  will  evidently  fulfil  the  conditions  required, 
y  being  here  greater  than  x  by  e.  And  if  y  be  taken  =ix — <?, 
or  x=y-\-e^  we  shall  obtain,  by  a  similar  substitution,  an 
equation  whose  roots  are  less  than  those  of  the  given  equation 
by  €. 

485.  Whence,  also,  as  e,  in  the  above  case,  is  indeterminate, 
this  mode  of  substitution  may  be  used  for  destroying  one  of  the 
terms  of  the  proposed  equation.  For  putting  in  the  above  ex- 
pression the  coefficient  —  m£?-}-A  =  o,  we  shall  have 

A         ,                           A 
e= — ,  and  a:=^y  —  e=:y : 

where  it  is  plain,  that  the  second  term  of  any  equation  may  he 

taken  away,  hy  substituting  for  the  unknown  quantity  some  other 

unknown  quantity,  together  with  such  a  part  of  the  coe^cient  of 

the  second  term,  taken  with  a  contrary  sign,  as  is  denoted  by  the 

index  of  the  highest  power  of  the  equation. 

Thus,  for  example,  to  transform  the  equation  a?^ — 9x'^-\-lx 

+  12z=0  into  one  which  shall  want  the  second  term.    Assume 

a?=y-|-3  ;  then 

a;3— y3_^_9y2_|_27y+27^ 
—  9a;2=     _9y2— 54y  — 81  (  _^ 
-{-Ix  =:  4- 7^4-21  (—"' 

+  12  =  •  +12) 

that  is,  y"^—  20y — 21  —0  ;  and  if  the  values  of  y  be  a,  b,  c,  the 

values  of  a;  are  a-4-3,  J+3,  and  c+3. 

The  third  term  of  the  proposed  equation  may  also  be  taken. 

away  by  means  of  the' coefficient,  or  formula, 

m(m — 1)  „      .         ,  \  J     .   Ti     rt 
-^- — ie^  —  {m—l)Ae+B=0, 
2 

29 


326  RESOLUTION  OF  EQUATIONS. 

where  the  determination  of  e  requires  the  solution  of  an  equa- 
tion of  the  second  degree  ;  and  so  on. 

486.  Any  proposed  equation  may  he  transformed,  into  another, 
the  roots  of  which  shall  he  any  multiples  or  parts  of  those  of  the 
former. 

Thus,  let  there  be  taken,  as  in  the  former  propositions)  the 
general  equation 

a'"  +  Aa:'"-i+Ba;'»-2+Ca?'"-3-f  .  .  Ta:+V=iO.  (1). 

And,  in  order  to  convert  it  into  another,  whose  roots  shall 
be  some  multiple  of  those  of  the  given  equation  ;  let  there  be 

V 
put  y  =  ex,  or  ac=-. 

Then,  by  substituting  this  value  for  x  in  the  proposed  equa- 
tion, there  will  arise 

r +Ay^+B?^,+  ....  T?^+V=0. 

gm     '         gm—l  gm-2  g 

And,  consequently,  if  this  be  multiplied  by  e*",  we  shall  have 

y"»4-Ae3/'"-^-fBe2y'"-i+  ....  Tc'»-iy+ Ve'"  =  0, 
which  equation  will  evidently  fulfil  the  conditions  required,  y 
being  equal  to  ex. 

And  if  y  be  put  =-,  or  x^ey^  we  shall  obtain,  by  a  similar 

€ 

substitution  of  this  value  for  a:,  and  then  dividing  by  e*",  the 
equation 

A  B  T       V 

^  ^  e^  e^^       ^  e^'-i^c'"        ' 

where  the  roots  are  equal  to  those  of  the  proposed  equation, 
divided  by  e. 

And  it  may  easily  be  proved,  that  if  the  alternate  terms, 
beginning  with  the  second,  he  changed,  the  signs  of  all  the  roots 
are  changed. 

487.  For  a  more  particular  account  of  the  general  Theory 
and  Doctrine  of  Equations,  see  Bonnycastle's  Algebra,  vol. 
ii.  8vo.  Bridge's  Equations,  and  Lagrange's  Traite  de  la  Re- 
solution des  Equations  Numeriques ;  where  the  intelligent 
reader  will  find  a  full  investigation  of  this  part  of  analysis. 

^  ii.  resolution  of  cubic  equations  by  the  rule  of 
Cardan,  or  of  Scipio  Ferreo. 

488.  Cubic  equations^  as   has  already  been  observed  in 


RESOLUTION  OF  EQUATIONS.  327 

Ghap.  VIIL,  are  of  two  kinds  ;  that  is,  pure  and  adfected.  All 
pure  equations  of  the  third  degree  are  comprehended  in  the 
formula  x^=:n,  where  n  maybe  any  number  whatever,  j90«- 
tive  or  negative,  integral  or  fractional.  And  the  value  of  x  is 
obtained  by  extracting  the  cube  root  of  the  number  n. 

489.  But  in  this  manner,  we  obtain  only  one  value  for  x ; 
whereas  every  equation  of  the  third  degree  has  three  values. 
In  order  to  show  how  the  two  remaining  values  of  x  may  be 
determined  in  equations  of  the  above  form,  let  us,  for  example, 
consider  the  equation  a;^  — 8  — 0;  where  a:  is  readily  found 
=2.  ■  And  as  2  is  a  root  of  the  proposed  equations,  it  is  plain 
that  x^  —  8  must  be  divisible  by  x—2  :  therefore,  this  division 
being  actually  performed,  the  quotient  will  be  a'-+2a;-h4. 

Hence  it  follows,  that  the  equation  x^ — 8  =  0,  may  he  re- 
presented by  these  factors  ; 

(x—2)x(x'^+2x-{-4)  =  0. 

490.  Now  the  question  is,  to  know  what  number  we  are  to 
substitute  instead  of  x,  in  order  that  a:^  — 8=i0  ;  and  it  is  evi- 
dent that  this  condition  is  answered  by  supposing  the  product 
which  we  have  just  found  equal  to  0 :  but  this  happens,  not 
only  when  the  first  factor  a:— 2=0,  which  gives  x=2,  but 
also  when  the  second  factor  x^-{-2x-{-4=:z0. 

Let  us,  therefore,  make  x'^-\-2x-\-4:=:0  ;  then  a:=  — Ij^ 
-^/  — 3.  So  that  besides  the  case  in  which  x=2,  we  find  two 
other  values  of  x,  which  will  satisfy  the  equation  x^  —  8=0. 
It  is  true,  as  Eulf,r  justly  observes,  that  these  values  are  im- 
aginary ;  but  yet  they  deserve  attention. 

491.  What  has  been  just  said  applies  in  general  to  every 
pure  cubic,  such  as  x^=:n,  and  the  three  roots  or  values  of  x, 
may  be  found  in  a  similar  manner.  To  abridge  the  calcula- 
tion, let  us  suppose  ^  7t=n\  so  that  'n=z7p  ;  the  proposed 
equation  will  then  assume  this  form,  x^ — n'^  =  0,  which,  be- 
ing divided  by  x—n,  will  give  for  the  quotient  x^-^-n'x-^-n'"^. 
Consequently,  the  equation  x? — w=0,  may  be  represented  by 
the  product  (a?— n')  (x''^-f-7i'.'r+n'-)  =  0,  which  is  in  fact  =0, 
not  only  when  a?  — /i'=i:0,  or  x=.n' ;  but  also  when  x^-\-n'x-\- 
n'2— -0.     Now  this  expression  contains  two  other  values  of  x, 

for  it  gives  xt=z ^ — a/~^  5  ho\h  of  which  answer  the 

2        2    V  • 

required  condition, 

492.  All  adfected  cubic  equations^  after  being  properly  re- 
duced by  the  known  rules,  may  be  exhibited  under  the  follow- 


328  RESOLUTION  OF  EQUATIONS. 

ing  general  forms  ;  namely,  x^-\-ax'^-\-bx  =  0,  and  x^-{-a'x^-}- 
b'x-i-cfz=0,  where  a,  b,  a',  b\  and  c\  may  be  any  numbers 
Vfh^lev ex,  positive  or  negative,  integral  or  fractional. 

493.  The  solution  of  a  cubic  equation,  of  the  form  x^-\-ax'^ 
+  bx~0,  is  attended  with  no  difficulty  ;  since  it  may  at  once 
be  put  under  the  form  xX[x'^-\-ax-\-b)  =  0  ;  and  it  is  evident 
that  the  product  a?  x(a;"+«ai-j-^)  may  be  =0,  in  two  ways, 
that  is,  when  a;  =  0,  or  x'^-^ax-\-bz=zQ  \  so  that  nothing  now 
remains,  but  to  find  the  values  of  x  in  the  quadratic  equation 

x'^-\-ax-\-b  —  Oy  which  are  readily  found  to  be  x= ±\^/{a'^ 

— 4b).  Consequently,  the  three  values  of  x,  which  answer 
the  required  condition,  are  0,— ^^-[-^-/(a^— 4Z>),  and  —^a~^ 

494.  An  adfected  cubic  equation  is  said  to  be  complete, 
when,  after  being  properly  reduced  by  the  known  rules,  it  is 
o[  the  form  x^-i-a'x'^-jrb'x-\-c'  =  0.  And  it  has  already  been 
shown,  that  every  cubic  equation  of  the  above  form,  whose 
roots  are  r\  r",  r",  may  be  transformed  into  another  deficient  in 
its  second  term,  by  substituting  y— Ja'  for  x  in  the  given  equa- 
tion ;-  in  which  case  the  roots  of  the  transformed  equation 
will  be  r— ia'  / — ^'  r"—-^a' ;  if,  therefore,  the  roots  of  the 
transformed  equation  be  known,  the  roots  of  the  given  equation 
will  be  known  also.  Hence  the  resolution  of  a  cubic  equation 
complete  in  all  its  terms  will  be  -effected,  if  we  can  arrive  at 
the  resolution  of  it  in  the  form  x'^-^ax^=.b.  In  which  a  and  b 
may  be  any  positive  or  negative  numbers  whatever. 

495.  For  this  purpose,  let  there  be  taken  x=.y-\-  z,  and  the 
above  equation,  by  substitution,  will  become  y^4-3y^^  +  3y^'^ 
-\-z'^-]rai/-]raz=ib. 

Or,  because  3i/^z-\'3i/z^r:z3i/z{y-{-z),a.r\dai/-\-az=za{i/-\-2)y 
it  will  be  y^-{-z''^-\-{3i/z-\-a){y-^z)=:b. 

Now,  as  another  unknown  quantity  has  been  introduced  into 
the  equation,  another  condition  may  be  annexed  to  its  solution. 

Let  this  condition  be,  that  3yz-\-a=:0,  or  z=——,  in  which 

case  the  transformed  equation  becomes 

y^'\'Z^=b,  or  by  substitution  y^  —  ———b  ; 

.*.  y^  —  by^—^a^ ;  which  equation  solved,  gives 

y—V  [^^+ v(i^^+iT^^)l  '  •■•  ^^"^®  z'^=:b—y'^,  we  have 

;.=^[i6-V'(i&2+^i_a^];  and  x=y+z=^\^b^^/(\b'^^ 


RESOLUTION  OF  EQUATIONS.  329 

where  by  taking  a  and  i  in  -f  or  —1,  as  they  may  happen 
to  be,  we  have  always  one  root  of  the  transformed  equation  ; 
and  this  is  the  formula  which  is  called  the  Rule  of  Cardan. 

496.  And  since  one  value  of  x  is  now  determined,  the  equa- 
tion may  be  depressed  to  a  quadratic,  from  which  the  other 
two  roots  may  be  readily  found. 

Ex.  1.  Given  x^  +  2xz=zl2,  to  find  the  values  of  a; 

Comparing  this  with  the  general  equation,  x^-\-axz=b,  we 
have  a  =  2,  and  b— 12  ;  therefore,  by  substituting  these  values 
for  a  and  b  in  the  above  formula  (I), 

x=l/  [GJrV[{^Q+M+V  [6-  V(36+^V)] 
=^  (6  +  6.024633)+^  (6  —  6.024633) 
=|/  (12.024633)  +  ;/  (  — .024633)zz:2.29  — .29=2. 

One  root  of  the  equation,  therefore,  is  2  ;  divide  x^-{-2x — 
12  by  x — 2,  and  the  quotient  is  aj^ — 2a:  +  6  ;  '.'.x^ — 2a:+6=0, 
whose  roots  are  Id:  V  — 5.  Hence,  the  three  roots  of  the 
equation  are  2,  1  +  -^/  — 5,  1 — ■\/  —  5,  the  two  last  of  which 
are  imaginary. 

Ex.  2.  Given  a;^— 48a;=128,  to  find  the  values  of  x. 

Here,  by  comparing  this  with  the  equation,  (Art.  494),  we 
have  a=— 48,  and  ^>=^128  ; 

.•.a;=^  [64+  ^(4096 -4096)]  +  ^  [64  —  -v/(4096-4096)] 
=^  (64  +  0)+^  (64-^0)=4+4  =  8. 

One  root  of  the  equation,  therefore,  is  8  ;  divide  x^ — 48a; 
— 128  by  y  — 8,  and  the  quotient  is  x"-\-Sx-{-\Q\  .-.x'^-^r^x-^ 
16  =  0,  whose  roots  are  —  4  +  0  ;  the  three  roots  of  the  pro- 
posed equation  are  8,  —4,  — 4,  the  two  last  of  which  are 
equal. 

497.  Hence  we  may  infer,  if  a  be  negative,  and  2Y<^^>  taken 
with  a  positive  sign,  equal  to  16^,  or  i&2_|__i_^3_-() .  then  two 
roots  of  the  proposed  equation  are  always  equal. 

498.  But  if  a  be  negative,  and  -j^^^,  taken  with  a  positive 
sign,  greater  than  ^6^  .  \\\^QXi  \b'^-\--^ja^  is  a  negative,  quantity  ; 
and  consequently,  ^/ {\i'^ -\- yjO?)  is  imaginary. 

Although  the  value  of  x  cannot  be  obtained  from  Cardan's 
formula,  (Art.  495),  by  the  ordinary  method,  we  are  not,  how- 
ever, to  conclude,  that  the  value  of  x,  in  this  case,  is  imagi- 
nary ;  since  it  may  be  proved  to-be  a  real  quantity  after  the 
following  manner. 

499.  For  this  purpose,  let  \b  be  represented  by  a\  and 
^/{\b'^•\•■^a'^),  supposed  imaginary,  by  b'-y/ ~\  ;  then  x=^^ 
(a^+5V-l)+^  (a'-^V-l)-  Now,  let  ^  {a' -^b' ^/ -~\) 
and  ^^  [a'  —  b' ■}/  —  \)  be  expanded  by  means  of  the  binomial 
theorem ;  and  since,  by  adding  the  resulting  series  together, 

29* 


330  RESOLUTION  OF  EQUATIONS. 

the  terms  involving  the  imaginary  quantity  -y/  — 1  destroy  one 
another,  we  shall  have 

4  h"^         105'*       1545's 

which  is  a  real  expression.  When  a'  is  greater  than  h'  ;  the 
above  series  converges  rapidly,  and  a  few  of  the  first  terms 
will  give  a  near  value  of  the  root  required.  But  if  a'  is  less 
than  h\  h'^  —  1  must  be  put  for  the  first  term  of  the  hinomialy 
and  a'  for  the  second  :   See  Clairaut's  Algebra,  Vol.  II. 

Ex.  3.  Given  a;^  — 6a;=:5.6,  to  find  the  values  of  x.     Com- 
paring this  with  the  equation  a;3+aa;=5,  we  have 
a=: — 6,  and  Z>z=5.6  ;  therefore, 
x=^  [2.8+ V(7.84-8)]-f^  [2.8- V(7-S4-8)] 
=^  (2.8-f  .4V-1)+^  (2.8-..4V-1.) 
Now,  by  comparing  this  value  of  x,  with  ^  (a''-|-5'y'  — l)-f- 
^  {of — y-y/  — 1),  we  have  a'=2.8,  and  h'=iA  ;  .-.  substituting 
these  values  for  a'  and  h'  in  the  above  formula  (2),  xz=2'^  2-8 

(^  +  7^  -  I49SS08'  '^'=)  =  ^•«2(1+  .00227-.00002, 
&c.)  =2.826345  nearly. 

Here,  three  terms  of  the  series  are  sufficient,  on  account  of 
its  converging  so  rapidly,  to  give  an  approximate  value  of  x, 
which  is  exact  enough  for  all  practical  purposes.  And,  in 
fact,  the  value  maybe  still  found  more  accurate  by  continuing 
the  serieS'to  five  or  six  terms. 

Ex.  4.  Given  ;?6—3;s4— 2^^  —  8=0,  to  find  the  values  of  z. 

Let  z^  =  x-{-l,  and  the  equation  will  be  transformed  into  x^ 
— 5a:=12  ;  .•.  since  «:= — 5,  and5=rl2. 

x:=^  [6  + V(36- W)]  +  i/  [6- V(36- W)] 
=^  (6  +  5.6009)+^  (6~5.6009)==2.26376  +  .73624=3. 
And,  consequently,  z'^=x-\-l=4:,  or  z=:^2. 
500.  Two  roots  of  the  proposed  equation,  therefore,  are  2' 
and  —2  ;  divide  z^  —  3z^—2z^—8  by  z'^—4:,  and  the  quotient 
is   z*-{-z^-{-2  ;  .•.;&*  +  ^^  +  2  =  0,  whose   roots    are    z  =  dtz 
■\/{ — J±  J-y/ — 7).     Hence  four  roots  of  the  proposed  equation 
are  imaginary. 

It  may  be  observed  that,  in  general,  all  equations,  as  z^"*-^ 
az^'"-{-bz'"-{-c=Oy  may  be  reduced  to  one  of  the  third  degree, 
by  putting  z"'=x—^a. 

Ex.  5.  Given  a;3  +  30a:=117,  to  find  the  values  of  aj. 

Ans.  x  =  3,  or  — IJ-J-/  — 3. 
Ex.  6.  Given  a:3+9a!:=270,  to  find  the  values  of  x. 

Ans.  x  =  Q,  or  — 3^6-/  — 1. 
Ex.  7.  Given  x^ — 36a;=91,to  find  the  values  of  ar. 

Ans.  a:=7,  or  -J  +  J-v/-^. 


RESOLUTION  OF  EQUATIONS.  331 

Ex.  8.  Given  x^  —  6x'^-i-l0x—8=0,  to  find  the  values  of  a?. 

Ans.  a;=:4,  or  l±-v/— I. 
Ex.  9.  Given  x^  —  Sx — 4  =  0,  to  find  the  values  of  a?. 
Ans.  a?=2.2;  1.1-4--/— .63  ;  — 11— -v/~-^3,  very  Ticar/y. 
Ex.  10.  Given  a;3-|-24a;=250,  to  find  the  value  of  a:. 

Ans.  3;= 5.05. 

Ex.  11.    Given  ^3— 602+13^—12=0,  to  find  the  values 

of  2.  Ans.  ;?=3,  or  — |jt:i\/  — 7. 

Ex.  12.  Given  2a;3  — 12x24-36a—44,  to  find  the  value  of  a:. 

Ans.  2.32748,  &c. 

§  III.    RESOLUTION  OF  BIQUADRATIC  EQUATIONS  BY  THE 
METHOD  OF  DeS  CaRTES. 

501.  The  same  observation  may  be  applied  to  biquadratic 
equations  as  was  applied  to  cubic  equations  in  (Art.  494),  that, 
since  the  equation  x^-\-a'x^-\-b'x'^-\-r'x-\-&'=.0,  may  be  trans- 
formed into  another  which  shall  be  deficient  in  its  second  term, 
and  whose  roots  shall  have  a  given  relation  to  the  roots  of  the 
given  equation,  the  complete  solution  of  a  biquadratic  equation 
will  be  eftected,  if  we  can  arrive  at  the  solution  of  it  in  the  form 

x^i-ax'2  +  bx+c  =  0     ..*..(!); 
where  a,  b,  c,  may  be  any  numbers  whatever,  positive  or  ne- 
gative. 

502.  In  the  solution  of  a  |j|^quadratic  equation,  after  the 
manner  of  Des  Cartes,  the  formula  x^-\-ax^-{-bx-{-cis  suppos- 
ed to  be  the  product  of  two  quadratic  factors,  x^-i-px-\-q  and 
x'^-{-rx-\-s,  in  which  p,  q,  r,  s,  are  unknown  quantities.  Or, 
which  is  the  same,  the  biquadratic  equation  x'^-i-ax^-\-bx'\-c=0 
is  considered  as  produced  by  the  multiplication  of  the  two 
quadratics, 

(2)  •.      .     .      .     x'^-hpx-\-q=0;  x^-{-rx-{-s=0    .     .     .    (3). 

503.  Hence,  by  the  actual  multiplication  of  the  above  two 
factors,  we  shall  have 

x'^-{-{p-\-r)x^-{-{s-\rq-hpr)x^-]r{ps-\-qr)x-\-qsz=: 
a;*  -\-ax^  -{-bx-\-c. 

And,  consequently,  by  equating  the  coefficients  of  the  like 
powers  of  a?  in  this  last  equation,  we  shall  have  the  four  fol- 
lowing equations,  , 
p-\-r=zO  ;  s-{-q-{-prz=za  ;  ps-\-qr-=ib  ;  qs=zc. 
Or,  if  — p,  which  is  the  value  of  r  in  the  first  of  these,  be 
substituted  for  r  in  the  second  and  third,  they  will  become, 

s-\-q=a+p'^  ;    s—q=-  ;  qs=c. 
*      p 

Whence,  subtracting  the  square  of  the  second  of  these  from 


332  RESOLUTION  OF  EQUATIONS. 

that  of  the  first,  and  then  changing  the  sides  of  the  equation, 
we  shall  have 

a2-j-2ap2_|_p4 _— 4^^j  or  4c. 

And,  therefore,  by  multiplying  by  "p^,  and  placing  the  terms 
according  to  the  order  of  their  powers,  the  result  will  give, 
;)«-f2a;>4+(a2— 4c)p2-&2.     ^     .     (4)^ 

From  which  last  equation,  if  there  be  put  p^=^,  we  shall 
have,;^3+2a;52^(a2^4c)^z=62 (5). 

Hence,  also,  since  ^+7=a+»2  and  s—q= — ,  there  will 

P 
arise,  by  addition  and  subtraction, 

^=la+ip24._.    q^la+lp'^-—; 

where  j9  being  known,  s  and  q  are  likewise  known. 

And,  consequently,  by  extracting  the  roots  of  the  two  as- 
sumed quadratics,  (2)  and  (3)  ;  or  of  their  equals,  x^-{-px-^ 
q—0,  and  x"^ — px 4- s z=zO  ;  we  shall  have 

«=-iP±V(ip'-«) (6); 

=o=},P±V{ip''-^)  •    • .•    ■    (7); 

which  expressions,  when  taken  in  +  and   — ,  give  the  four 
roots  of  the  proposed  biquadratic,  as  was  required. 

504.  It  may  be  observed,  that  whichever  of  the  values  of 
the  unknown  quantity,  in  the  fpbic  or  reduced  equation  (5), 
be  used,  the  same  values  of  x  will  be  obtained. 

505.  To  this  we  may  further  add,  that  when  the  roots  of 
the  cubic,  or  reduced  equation  (5),  are  all  real,  then  the  roots 
of  the  proposed  biquadratic  are  all  real  also.  But  if  only  one 
root  of  the  cubic  equation  (1)  be  real,  and,  therefore,  the  other 
tivo  imaginary ;  then  the  proposed  biquadratic  will  have  two 
real  and  two  imaginary  roots. 

Ex.  1.  Given  the  equation  a:"^  — 3a;2-}-6a?4-8  =  0,  to  find  its 
roots,  or  the  values  of  x. 

Comparing  this  equation  with  a;*4-«^^+^^+c  =  0j  we  have 
a=  — 3,  6  =  6,  and  c  —  Q  ;  therefore, 

23+20224- (fl2_4c)^_52^^3_622_|.23^  — 36  =  0. 

Let  2=y+2,  and  substitute  y+2  for  z  in  the  latter  equa- 
tion ;  the»resulting  equation  is  y^  —  3by  —  ^S  =  0.     Now,  by 
comparing  this  last  equation  with  x'^'\-ax=zb,  we  have  a=  — 
35,  and  ^>  =  98  ;  therefore,  (Art.  495), 
y=^  [49+iVl65856)]+|/  [49-1^(65856)] 

=^  (49  +  28.514)-!-  ^  (49-28.514)=^  (77.514)+^  20. 
466) 
=4.264+2.736=7. 


RESOLUTION  OF  EQUATIONS.  333 

Hence,  z—y-\-2  —  9,  and  p^=z=9,  or  p=  ±3  ; 
.-.(Art.    504),  taking  p:=3,  s=  —  f +f +1==3  +  1=4,  and 
5"=— 14-|— 1=2.     Consequently,  by  substituting  these  va- 
lues for p,  q,  and  s,  in  the  equations  (2),  (3),  we  shall  have 
x2+3a:4-2  =  0,  and  a:2  — 3a;-h4^0; 
.-.  a;=  — |±i,  and  a^^fij^— 7  ; 
so  that  the  four  roots  of  the  given  equation  are  —1,  —2,  |-|- 

iV-7,  ^-W-^- 

Ex.  2.  Given  x^  —  6x^—l7x-\-2l  =0,  to  find  the  values  of  x. 

Ans.  a::=3,  or  1  ;  or  — 2±'\/  — 3. 

Ex.  3,   Given  the  equation  a:*  — 4a;3  — 8x4-32=0,  to  find  its 

roots,  or  the  values  of  x.  Ans.  4,  or  2  ;  or  —  l±y — 3. 

Ex.  4.   Given  the  equation  x'^—6x^-\^3x^-\-2x—\0  —  0,  to 

find  its  roots,  or  the  values  of  x. 

Ans.  —1,  or  +5  ;  or  IrtV  — 1- 

Ex.  5.  Given  a;*  —  9a:''+30a;2— 46a:  +  24=0,   to   find  the 

roots,  or  values  of  a;.  Ans.  a^^l,  or  4  ;  2  j^-y/— 2. 

Ex.  6.  Given  a;*H-16a;34-99x2-|-228a:4- 144  =  0,  to  find  the 

roots,  or  values  of  x. 

Ans.  a:=  — 1,  —3  ;  or  — Gi^  — 12. 
Ex.  7.  What  two  numbers  are  those,  whos^  product,  mul- 
tiplied by  the  greater,  is  equal  to  1  ;  and  if  from  the  square 
of  the  greater,  added  to  six  times  the  lesser,  the  cube  of  the 
lesser  be  subtracted,  the  remainder  shall  be  8. 

Ans.  -V2iV(l  +  V2),  +V2iV(l-V2). 

§  IV.    RESOLUTION    OF  NUMERAL  EQUATIONS  BY  THE  METHOD 
OF  DIVISORS. 

506.  Since  the  last  term  (v)  of  the  equation  {a)z=zx"'-\- 
Ax'^—^-i-Bx'^—^  ....  Ta;+v=o,  is  equal  to  the  product 
of  all  its  roots,  it  is  evident,  that  if  any  of  those  roots  be  whole 
numbers,  they  will  be  found  among  the  divisors  of  that  term. 
To  discover,  therefore,  whether  any  of  the  roots  of  a  given 
equation  be  whole  numbers,  we  have  only  to  find  all  the  divi- 
sors of  its  last  term,  and  substitute  each  of  them,  first  with  the 
sign  -}-  3-i^d  then  with  the  sign  — ,  for  x,  in  the  given  equa- 
tion, such  of  them  as  reduce  the  equation  to  0=0,  will  be 
roots  of  the  equation. 

507.  Or,  if  the  divisors  of  the  last  term  should  be  too  nu- 
merous, the  equation  may  be  transformed  into  another,  that 
shall  have  its  last  term  less  than  that  of  the  former  ;  which  is 
done  by  increasing  or  diminishing  the  roots  by  1,  or  some 
other  quantity. 

Ex.  1.  Given  ir^— a;^— 2a;  +  8r=:0,  to  find  the  roots  of  the 
equation,  or  values  of  x. 


334  RESOLUTION  OF  EQUATIONS. 

Here  the  divisors  of  its  last  term,  are  1,  2,  4,  8  ;  substitute 
1,  2,  4,  8,  and  — 1,  —2,  —  4,  — 8,  for  x  in  the  given  equation, 
and  —2  will  be  found  to  be  the  only  one  of  these  numbers 
which  gives  the  result  0  ;  — 2  therefore  is  the  only  integral 
root  of  the  equation.  Hence,  oc-\-2  will  divide  x'^—x'^—2x-{- 
8  without  a  remainder  ;  let  this  division  be  made,  and  the 
quotient  being  put  equal  to  0,  we  shall  have  x^ — 3a?+4  =  0,  a 
quadratic  equation  which  contains  the  other  two  roots.  The 
solution  of  this  quadratic  gives  xz=.^J^^y^^7  ;  the  three 
roots  of  the   given   equation,  therefore,  are  ~2,  f-hiV — 7, 

508.  The  integral  roots  of  any  numeral  equation  of  the 
Ivind  above  mentioned,  may  also  be  found,  by  Newton's  Me- 
thod of  Divisors,  which  is  founded  upon  the  following  prin- 
ciples. 

Let  one  of  the  roots  of  the  equation  (a)  =  0,  be  — a,  or, 
which  is  the  same,  let  the  proposed  equation  be  represented 
under  the  form  (a'+«)p=0,  where  the  binomial  x-\-a  denotes 
one  of  the  divisors,  or  factors,  of  which  the  equation  is  com- 
posed, and  p  the  product  of  the  rest.  Then,  if  three  or  more 
terms  of  the  arithmetical  series,  2,  1,  0,  —1,  — 2,  be  succes- 
sively substituted  for  x,  the  divisors  of  the  results,  thus  ob- 
tained, will  be 

a-\-2,  a+1,  a,  a  —  1,  and  a—^. 

Arid  as  these  are  also  in  arithmetical  progression,  it  is  plain 
that  the  roots  of  the  given  equation,  when  integral,  will  be 
some  of  the  numbers  in  such  a  series. 

Whence,  if  a  progression  of  this  kind,  whose  common  dif- 
ference is  1,  can  be  found  among  the  divisors  of  the  results 
above  mentioned,  by  taking  one  number  out  of  each  of  the 
lines,  that  term  of  it  which  answers  to  the  substitution  of  0 
for  X,  taken  in  +  or  — »  according  as  the  series  is  increasing 
or  decreasing,  will  generally  be  a  root  of  the  equation. 

Ex.  2.  Givena;'^H-a;*  — 14a;3  — 6a;2-f20a:+48  =  0,  tofindthe 
roots  of  the  equation,  or  values  of  x. 
Divisors. 
1,2,  5,  10,  25,  50,  1 

1,  2,  3,  4,  6,  8,  12,  24,  48,     2 
1,2,  3,  4,  6,  9,  12,  18,  36,     3 

Here  the  numbers  to  be  tried  are  2,  3,  —4,  all  of  which 
are  found  to  succeed ;  so  that  the  equation  has  three  integral 
roots  ;  namely,  2,  3,  —4.  The  equation  whose  roots  are  2, 
3,  —4,  is  {x-2)  .  (x-S)  .  {x-{-4)  =  x^--x^  —  \4x-\-24==0Jet 
the  given  equation  be  divided  by  it,  and  the  quotient  is  x'^-{- 
2<r4-2=:0,  whose  roots  are  — lzt:/~l  5    ^^®  ^^®    ^^^^^   ^^ 


Num. 

Results. 

1 

50 

0 

49 

—  1 

36 

Progress, 


RESOLUTION  OF  EQUATIONS. 


335 


the  proposed  equation  are,  therefore,  2,  3,  —4,  —  l  +  i/— 1» 

509.  If  the  highest  power  of  the  unknown  quantity  has  any- 
coefficient  prefixed  to  it,  let  the  equation  be  assumed  of  the  form 
(nx-{-a)p  =  0,  and  substitute  2,  1,  0,  —1,  —2,  successively  for 
X,  as  in  the  former  instance. 

Then,  as  before,  the  divisors  of  the  several  results,  arising 
from  this  substitution,  will  be  the  terms  of  the  arithmetical 
series, 

2n  +  a,  n-\-a,  a,  — 7i+a,  and  — 2n+a; 
where  the  common  difference  n  must  be  a  divisor  of  the  first 
term  of  the  equation,  or  otherwise  the  operation  would  not 
succeed. 

Hence,  in  this  instance,  the  progressions  must  be  so  taken 
out  of  the  divisors,  that  their  terms  shall  differ  from  each  other 
by  some  aliquot  part  of  the  coefficient  of  the  first  term. 

Therefore,  if  the  terms  of  these  series,  standing  opposite 
to  0,  be  divided  by  the  common  difference,  the  quotient  thus 
arising,  taken  in  -j-  and  — ,  according  as  the  progression  is 
increasing  or  decreasing,  will,  generally  be  the  roots  of  the 
equation. 

It  is  necessary  to  continue  the  series  2,  1,  0,  —1,  —2,  far 
enough  to  show  whether  the  corresponding  progression  may 
not  break  off,  after  a  certain  number  of  terms  ;  which  it  never 
can  do  when  it  contains  a  real  rational  root. 

Ex.  3.  Given  2a;3  —  3a;2+ 16a: —24  =  0,  to  find  the  roots  of 
the  equation  or  values  of  x. 

2,  successively,  for  a?,  as  in  the 


Substituting  2,  1,  0,  —  1, 
former  case,  we  shall  have 


Num. 

2 

1 

0 

—  1 

-2 


Results. 

12 

—   9 

-24 
-45 

—84 


Divisors. 
2,3,4,  6.,  12, 

3,  9, 

2,3,4,6,  8,  &c. 
3,5,9,15,45, 


Prog. 
—  1 
+  1 

+  3 
+  5 
+  7 


_.       1,2,3,4,    6,    7,  &c.   ^     .  . 

Where  the  progression  is  ascending,  the  number  to  be  tried 
is,  therefore,  |,'  which  is  found  to  be  a  root  of  the  equation. 

Let  the  given  equation  be  divided  by  a:  — f,  and  the  quotient 
is  2.r2  — 16  =  0,  whose  roots  are  db 2-^/2  ;  the  three  roots  of  the 
proposed  equation  are,  therefore,  — #,  +2^2,  — 2-/2. 

Ex.  4.  Given  a;*-|-a;3— 29a:2  — 9a;+180nr0,  to  fixid  the  roots 
of  the  equation.  Ans.  3,  4,  —3,  and  — 5. 

Ex.  5.  Given  a;*— 4a:3  — 8a;H-32  =  0,  to  find  the  roots  of  the 
equation,  or  values  of  x. 

Ans.  a? =2,  or  4  ;  or--l±v'--3. 


336  RESOLUTION  OF  EQUATIONS 

Ex.  6.  Given  a;3_5a;24.i0a:— 8=0,  to  find  the  integral  root 
of  the  equation.  Ans.  2. 

Ex.  7.  Given  a:*— •8a:3-|-a;2+82a;— 60  =  0,tofindthe  integral 
roots  of  the  equation.  Ans.  5,  and  —3. 

Ex.  8.  Given  a;^  — 9a:3-f  8a?2— 72=0,  to  find  the  roots  of  the 
equation,  or  values  of  x. 

Ans.  a:=— 3,  or  —2,  or  3  ;  or  liy--3. 


§  V.    RESOLUTION  OF  EQUATIONS  BY  NeWTON's  METHOD  OF 

Approximation. 

510.  The  methods  laid  down  in  the  preceding  section,  will 
be  found  suflicient  for  determining  the  integral  or  rational 
roots  of  equations  of  all  orders  ;  but  when  the  roots  are  irra- 
tional, recourse  must  be  had  to  a  different  process,  as  they 
can  then  be  obtained  only  by  approximation  ;  that  is  to  say, 
by  methods  which  are  continually  bringing  us  nearer  to  the 
true  value,  till  at  last  the  error  being  very  small,  it  may  be 
neglected. 

511.  Different  methods  of  this  kind  have  been  proposed, 
the  simplest  and  most  useful  of  which,  as  Lagrange  justly  re- 
marks, is  that  of  Newton,  first  published  in  Wallis's  Algebra, 
and  afterwards  at  the  beginning  of  his  Fluxions — or  rather 
the  improved  form  of  it,  given  by  Raphson,  in  his  works,  en- 
titled Analysis  jEquationem  U?iiversalis. 

512.  In  order  to  investigate  the  above-mentioned  method, 
let  there  be  taken  the  following  general  equation, 

x*^-]-px^—^-^qx"'—^-{-rx'^—^-{-  .  .  sx^-\-tx-\-u  =  0  .  (1). 
Then,  supposing  a  to  be  a  near  value  of  a;,  found  by  tiial,  and 
z  to  be  the  remaining  part  of  the  root,  we  shall  have  x=a-\-z  ; 
and,  consequently,  by  substituting  this  value  for  x  in  the  given 
equation,  there  will  arise 

(rt+;^)'"-fp(a+^)'"-i+  .  .  .  s{a-\-  zf  +  t{a-\-z)  +  uz=zO  ; 
which  last  •xpression,  by  involving  its  terms,  and  taking  the 
result  in  an  inverse  order,  may  be  transformed  into  the  equa- 
tion 

P  +  Q;j+R^2_|_s^34.  .  .  .  +2'-=0  .  .  (2), 
where  P,  Q,  R,  &c.  are  polynomials,  composed  of  certain  func- 
tions, of  the  known  quantities,  a,  m,p,  q,  r,  Sic.  which  are  de- 
rived from  each  other,  according  to  a  regular  law. 

513.  Thus,  by  actually  performing  the  operations  above  in- 
dicated, it  will  be  found  that 


RESOLUTION  OF  EQUATIONS.  337 

wliich  value  is  obtained  by  barely  substituting  a  for  a?  in  the 
equation  first  proposed. 

And,  by  collecting  the  several  terms  of  the  coefficients  of  z, 
it  will  likewise  appear,  that 

Q,=ma^-^-^m{m  —  \)pa'^-^-{-  ...  +2sa+t; 
which  last  value  is  found  by  multiplying  each  of  the  terms  of 
the  former  by  the  index  of  a  in  that  term,  and  diminishing  the 
same  index  by  unity. 

514.  Hence,  since  z  in  equation  (2)  is,  by  hypothesis,  a 
proper  fraction,  if  the  terms  that  involve  its  several  powers 
z"^,  z^,  ST"*,  &c.  which  are  all,  successively,  less  than  z,  be  neg- 
lected in  the  transformed  equation,  we  shall  have 


P+Q^=0,  or 


a"'-\-pa^—^-\- -\-ta^u 


And,  consequently,  if  the  numeral  value  of  this  expression 
be  calculated  to  one  or  two  places  of  decimals,  and  put  equal 
to  b,  the  first  approximate  part  of  the*  root  will  be  z=ib,  or 
x=a-\-b^=:a'. 

Whence  also,  if  this  value  of  a?,  which  is  nearer  its  true 
value  than  the  assumed  number  a,  be  substituted  in  the  place 
of  a  in  the  above  formula,  it  will  become 

_      a'"'+pa^'"~^+  •  •  •  •  -\-ta'-\-u 

~     ma^"'—^-{-{m  —  l)pa''^-'^-{-.-\-t 

which  expression  being  now  calculated  to  three  or  four  places 

of  decimals,  and  put  equal  to  c,  we  shall  have,  for  a  second 

approximation  towards  the  unknown  part  of  the  root, 

z=:c,  or  x=a'-{-c=a". 

And,  by  proceeding  in  this  manner,  the  approximation  may 
be  carried  on  to  any  assigned  degree  of  exactness  ;  observing 
to  take  the  assumed  root  a  in  defect  or  excess,  according  as  it 
approaches  nearest  to  the  root  sought,  and  adding  or  subtract 
ing  the  corrections  6,  c,  &c.  as  the  case  may  require. 

515.  A  negative  root  of  any  equation  may  also  be  found  in 
the  same  manner,  by  first  changing  the  signs  of  all  the  alter- 
nate terms,  and  then  taking  the  positive  root  of  this  equation, 
when  determined  as  above,  for  the  negative  root  of  the  propos- 
ed equation. 

516.  In  the  practical  application  of  this  rule  we  must  en- 
deavour to  find  two  whole  numbers,  between  which  some  one 
root  of  the  given  equation  lies  ;  and  by  substituting  each  of 
them  for  x  in  the  given  equation,  and  then  observing  which  of 
them  gives  a  result  most  nearly  equal  to  0,  we  shall  ascertain 
the  whole  number  to  which  a?  most  nearly  approaches ;  we 

30 


338  R4i:S0LUTI0N  OF  EQUATIONS. 

must  then  assume  a  equal  to  one  of  the  whole  numbers  thus 
found,  or  to  some  decimal  number  which  lies  between  them, 
according  to  the  circumstances  of  the  case. 

517.  Since  any  quantity,  which  from  positive  becomes  ne- 
gative passes  through  0,  if  any  two  vvhol^,  numbers  n  and  n' ; 
one  of  which,  when  siabstituted  for  x  in  the  proposed  equation, 
gives  a.  positive,  and  the  other  a  negative  result ;  one  root  of  the 
equation  will,  therefore,  lie  between  n  and  n'.  This,  of  course, 
goes  upon  the  supposition  that  the  equation  contains  at  least 
one  real  root. 

518.  It  is  necessary  to  observe,  that,  when  a  is  a  'much 
nearer  approximation  to  one  root  of  the  given  equation  than 
to  an)*  other,  then  the  foregoing  method  of  approximation  can 
only  be  applied  with  any  degree  of  accuracy.  To  this  we 
also  farther  add,  that,  when  some  of  the  roots  are  nearly 
equal,  or  differ  from  each  other  by  less  than  unity,  they  may 
be  passed  over  without  being  perceived,  and  by  that  means 
render  the  process  illusory  ;  which  circumstance  has  been 
particularly  noticed  by  Lagrange,  who  has  given  a  new  and 
improved  method  of  approximation,  in  his  Traite  de  la  Reso- 
lution des  Equations  Numeriques.  See,  for  farther  particulars 
relating  to  this,  and  other  methods,  Bonnycastle's  Algebra, 
or  Bridge's  Equations. 

Ex.  1.  Given  a;^4-2a:2  — 8a7=r24,  to  find  the  value  of  x  by 
approximation. 

Here,  by  substituting  0,  1,  2,  3,  4,  successively  for  x  in  the 
given  equation,  we  find  that  one  root  of  the  equation  lies 
between  3  and  4,  and  is  evidently  very  nearly  equal  to  3 
Therefore  let  a=3,  and  x=:a-{-z. 

Cx^  =  a^+3a'^z-[-3az^+z^^ 
Then  ?  2x^=2a'^-\-4az-{-2z^         >  =24. 
(  —8x=—8a  —  8z  } 

And  by  rejecting  the  terms  z^-\-3az'^-{-2z^,  (Art.  514),  as  be- 
ing small  in  comparison  with  z,  we  shall  have 

a^-^2a'^  —  8a-\-3a'^z+4az-'8z=24:  ; 
24— a^— 2a2+8a      3 

•••^= — ^-rrz — ^=^=-09  ; 
3a^-t-4a  —  8  31 

and  consequently  x—a-{-2=:3.09,  nearly 

Again,  if  3.09  be  substituted  for  a,  in  the  last  equation,  we 
shall  have 

_24-a^—2a^-\-8a  _  24—29.503629  —  19.0962  +  24.72 

^-       3a2+4a— 8       ""  28.6443-f  12.36-8 

=  .00364  ;  and,  consequently,  x=:a  +  z=3.09  +  .00364  == 
3.09364,  for  a  second  approximation 


INDETERMINATE  COEFFICIENTS.        339 

And,  if  the  first  four  figures,  3.093,  of  this  number,  be  sub- 
stituted for  a  in  the  same  equation,  an  approximate  value  of  x 
will  be  obtained  to  six  or  seven  places  of  decimals.  And  by 
proceeding  in  the  same   manner  the  root  may  be  found  still 

more  correctly.  ^    ,  ,         ,        r    i 

Ex    2    Given  3a;5  +  4a;3— 5a;r=140,tofindthevalue  ofa:by 

approximation.  ^    Ans.  .=2.07264 

Ex  3  Given  a:*-9a;3  +  8a;2-3x+4  =  0,  to  find  the  value  of 
X  by  approximation.  Ans.  a:^  1.1 14789. 

Ex.  4.  Given  a:3  +  23  3a:2-39x-93.3=0,  to  find  the  va- 
lues of  X  by  approximation. 

Ans.  a:=2.782;  or  -1.36;  or  -24.72  ;  very  nearly. 

Ex  5  Find  an  approximate  value  of  one  root  of  the  equa- 
tion a:3+.x2  +  ^^90.  Ans.  a:=.4.10283. 

Ex.  6.  Given  a;3+6.75a:2-f-4.5a:-10.25=0,  to  hnd  the  va- 
lues of  X  by  approximation. 

Ans.  a;z=:.90018  ;  or  -2.023  ;  or  -5.627  ;  very  nearly 


CHAPTER  XVL 


ON 


INDETERMINATE  COEFFICIENTS,  VANISHING  FRAC- 
TIONS, AND  FIGURATE  AND  POLYGONAL  NUMBERS. 

§  I.    ON   INDETERMINATE   COEFFICIENTS. 

519.  This  is  a  species  of  investigation,  which  is  frequently 
used  for  obtaining  the  development  of  certain  fractional  and 
other  expressions,  without  having  recourse  to  the  operations 
of  division,  or  the  extraction  of  roots  ;  the  method  of  per- 
forming which  is  as  follows  : 

RULE. 

Assume  a  series,  or  other  expression,  with  unknown  coeffi 
cients,  for  that  which  is  required  to  be  found  ;  then,  having 
multiplied  it  by  the  denominator  of  the  given  fraction,  or 
raised  it  to  its  proper  powers,  find  the  value  of  each  of  these 


240         INDETERMINATE  COEFFICIENTS. 


coefficients,  by  equating  the  homologous  terms  of  the  two  ex- 
pressions, or  putting  such  of  them  as  have  no  corresponding 
terms,  equal  to  0,  as  the  case  may  require. 

Example  1.  Let  it  be  required  to  find  the  development  of 

,  according  to  the  above  method. 


Assume 


,  =  A  +  Ba:+Ca:2+Da:3  +  Ea;*,  &c. 


Then,  multiplying  the  right  hand  side   of  the   equation  by 
a'-^-Vx^^  and,  transposing  a,  we  shall  have 


0  =  Aa'-}-Ba' 


x^,  &;c. 


x-\-Ca' 
+  B6' 

And  by  putting  the  first  term,  and  the  coefficients  of  the 
several  powers  of  x,  each  =0,  there  will  arise  the  following 
equations : 


Hence, 


Aa^ —  a=0 
Ba^+AZ>'=0 
Ca'-i-Bb'  =  0 

Da'+C6'=0 

&c. 
a  a 

a'-\-b'x'^     a' 


or 


A=r 


B  =  -— A 

a 

a 
&c. 


■A  a; -Bx"^ 

a 


y 


-Ca:3,  &c 


Where  it  is  obvious,  that  each  coefficient,  in  parting  from  the 
second  inclusively,  is  equal  to  that  which  precedes  it,  multi- 

7/ 

plied  by —  :  which  law  renders  it  unnecessary  to  take  a 

greater  number  of  equations,  or  to  push  the  calculation  far- 
ther. 

Ex.  2.  Required  the  development  of 


ing  to  the  same  method. 
Assume 


a'-i-b^x-\-c'x^ 


accord- 


A  +  Bx-^Cx^-hDoc^  &c. 


a'-\-b'x-{-c'x^ 

Then  multiplying  the  right  hand  side  of  the   equation  by 
a^'\-b'x-\-c^x^,  and  transposing  a-{-bx,  we  shall  have 


=Aa'4-Ba' 

x+Ca' 

0:2+ Da' 

-    a-\-Ab' 

+  By 

+  Cb' 

-    b 

+  A</ 

+  Bc' 

INDETERMINATE  COEFFICIENTS. 


341 


And  by  putting  the  coefficients  of  the  several  powers  of 
a?=0,  there  will  arise  the  equations 


ka'—a=0 

Ca'+B6'+Ac'=0 

Da'+Cy+Bc^=0 
&c. 

Whence 


or 


A  = 
B  = 
C  = 


a 
a' 

a  a 


,b-Va 

a'  a 

c  a 

&c. 


a-\-hx 


a'-\-b'x-\-c'x'^ 

a         \a  a  /        \a  a      /  \a  a      / 

Where  each  coefficient,  in  parting  from  the  third  inclusively 
may  be  readily  deduced  from  the  two  that  precede  it.  So 
that  if  P,  Q,  R,  be  any  three  consecutive  coefficients,  we 
shall  have 

Ra'+Q6'+Pc'=0;  orR  =  — — Q— ^P.      * 

a  a 

Ex.  3.  Given  {x^+p)'^—(qx+rf=:x^-\-ax^  +  bx+c,io  find 
the  indefinite  coefficients  p,  q,  and  r. 

Here,  by  squaring  the  terms  on  the  left  hand  side  of  the 
equation,  and  collecting  those  that  are  alike,  we  have 

aj*4-(2;)  — 9^)aj2 — 2rqx-{-p^  —  r'^=x'^-{-ax^~\'bx-\-c. 
And  consequently,  by  equating  the  homologous  terms, 


2p-q^  =  a 

2p—a  —  q^ 

~2qr=b 

or 

-b=2qr 

p^^r^=:zc 

f^c^.f' 

Where  it  is  plain,  that  the  product  of  the  first  and  third  of 
these  equations  is  equal  to  \  of  the  square  of  the  second  ;  or 

Hence  the  value  of  p  may  be   found  by  a  cubic   equation, 
and  then  q  and  r  from  the  former  equations. 

A 

Ex.  4.  It  is  required  to  convert into  a  series  by  the 

above  method. 

Ans.  _(l +_+_-+— +-^,  +,  &c.) 
30* 


342  VANISHING  FRACTIONS. 

14- 2a: 
Ex.  5.  It  is  required  to  convert into  a  series  by 

the  same  method. 

Ans.  l  +  3x+4:x'^+7x^+llx^+18x^+  Slc. 

I ^ 

Ex.  6.  It  is  required  to  convert :; — — -  into  a  series 

1— 2a: — 3a;2 

by  the  same  method. 

Ans.  l+a;4-5a^2+13a:3-f41a;*+121a;54.  &c. 
Ex.  7.  It  is  required  to  convert  -^/(l— a:)  into  a  series  by 
the  same  method. 

"^'  2~274     2X6     2X6^~2lT678.T0~       • 


}  II.    ON  VANISHING    FRACTIONS. 

520.  Vanishing  fractions,  and  other  similar  expressions, 
are  such,  as  in  certain  cases,  become  equal  to  - ;  which  sym- 
bol, though  apparently  nugatory,  or  of  no  value,  must  not  be 
rejected  as  useless,  being  of  frequent  occurrence  in  several 
Algebraical  and  Fluxional  investigations,  where  it  will  often 
from  the  nature  of  the  subject,  denote  some  fixed,  or  determi- 
nate quantity. 

Thus,  if  a  be  made  to  represent  the  first  term  of  any  regular 
geometrical  series,  r  the  ratio,  and  n  the  number  of  terms,  we 
shall  have 

—=a-rar'^-\-ar^-\-ar^-\- ar'^^-\-ar'^^. 

r — 1 

Where  the  left  hand  member  of  the  equation  is  a  universal 
expression  for  the  sum  (S)  of  the  series,  whatever  may  be  the 
values  o(  a,  r,  and  n  ;  as  will  appear  by  dividing  the  numera- 
tor by  the  denominator. 

Let,  therefore,  the  ratio  or  multiplier  r,  be  taken  =1  ;  and 
the  expression  for  the  same  will  be 
a— a     0 

^=i~I=o- 

But  when  r=l,  the  original  series  becomes  of  the  form 

S  =  a-^a-\-a-\-a-\-  &c to  ;i  terms  ;  of  which  the  sum 

is,  evidently,  =zna ;  and,  therefore,  in  this  case,  it  follows,  that 

0  •• 

621.  And  in  the  same  way  it  might  be  shown,  that  this 
symbol  is  the  representative  of  various  other  quantities,  accord- 


VANISHING  FRACTIONS.  '       343 

ing  to  the  nature  of  the  expression  from  which  it  is  derived  ; 
but  it  will  be  here  sufficient  to  observe,  that  the  true  value  of 
any  fractional  expression  of  this  kind  may  be  readily  obtained 
as  follows. 

RULE. 

1.  If  both  the  terras  of  the  given  fraction  be  rational,  divide 
each  of  them  by  their  greatest  common  measure  ;  then,  if  the 
hypothesis  which  is  found  to  reduce  the  original  expression 

to  the  form  -,  be  applied  to  the  result,  it  will  give  the  true  va- 
lue of  the  fraction  in  the  state  under  consideration. 

2.  Where  any  part  of  the  fraction  is  irrational,  observe  what 
the  unknown  quantity  is  equal  to  when  the  numerator  and  de- 
nominator both  vanish,  and  put  it  =z  that  quantity  +  or  i ;  then, 
if  this  be  substituted  for  the  unknown  quantity,  and  the  roots 
of  the  surds  be  extracted,  to  a  sufficient  number  of  places, 
the  resuh,  when  i  is  put  =0,  will  give  the  true  value  of  the 
fraction. 

Example  1.  It  is  required  to  find  the  value  of  the  fraction 

,  when  X  is  equal  to  a. 

x  —  a 

cp'—d?-     0 
Here,  if  we  put  x=a,  there  will  arise =-.     But,  by 

division, ■=^x-\-a\  and  if  x  be  now  put  =a,  we  shall 

x—a 

a2 fj2  0 

have z=L2a  ;  whence  -,  or  the  given  fraction,  in  its  va- 
nishing state,  is  =2a. 

Ex.  2.  It  is  required  to  find  the  value  of  the  expression 

b(x—-Jax)      .  .  . 

^= *-^ ,  when  X  is  equal  to  a. 

X  —  a 

Here,  if  x  be  taken  r=a+i,  according  to  the  rule,  we  shall 

have  y=— t- ■ — -.     And,  by  extractmg  the  square 

root  of  a^  +ai,  and  then  dividing  by  i. 

Whence,  putting  the  indeterminate  quantity  «=0,  there  will 
arise 

y=V>; 


344  VANISHING  FRACTIONS. 

whicli  is  the  true  value  of  the  expression,  in  the  case  pro- 
posed. 

Ex.  3.  Let  there  be  taken,  as  another  example  of  this  kind, 
the  equation 
•  _P(a:— g)" 

where  P  and  Q  are  supposed  to  be  certain  functions,  or  com- 
binations of  X,  which  do  not  become  0  for  the  same  value  of  x. 
Then  taking  x  =  a,  the  expression,  according  to  this  hypo- 
thesis, will  become  of  the  form 

P_xO__0 
QxO"0" 
But  by  considering  the  indices  wi,  n,  of  the  proposed  frac- 
tion, under  each  of  the  relations 

m'^n,  m  =  n,  m<^n, 
we  shall  have,  by  division,  the  three  following  results  : 

_P(a: -«)>»-"     _P P_ 

^~         Q         '^"Q'^-QCo^-a)"— * 
And  consequently,  by  now  taking  x-=a,  there  will  arise 
_PX0      _P  P 

^~~Q~'^~Q'^""QX0' 

"Whence,  the  value  of  the  symbol  --,  in  this  case,  will  be  no- 
thing, finite,  or  infinite,  according  to  the  conditions  above 
mentioned. 

Ex.  4.  It  is   required   to   find  the   value  of  the    fraction 

*  when  X  is  equal  to  1.  Ans.  4. 

X  — X 

Ex.  5.  It  is  required  to  find  the  value   of    the  fraction 

»P«» fi"* 

,  when  xz=:a.  Ans.  ma"^—^. 


Ex.  6.  It  is  required  to  find  the    value  of  the    fraction 
when  X  is  equal  to  a.  Ans.  Sa^. 


x—a 


*.The  value  of  this  fraction  was  the  cause  of  a  violent  controversy  between 
Waring  and  Powell,  in  1760,  when  these  gentlemen  were  candidates  for  the 
mathematical  professorship  at  Cambridge  ;  Waring  maintaining  that  the  value 

X — x^ 
of  the  fraction  -rzi —  ^^  equal  to  4  when  a:=l,  and  Powell,  (or  rather  Maseres, 

who  is  commonly  thought  to  have  conducted  the  dispute,)  that  it  was  equal 
to-O. 

The  idea  of  vanishing  fractions  first  originated  about  the  year  1702,  in  a 
contest  between  Varignon  and  Rolle,  two  French  mathematicians  of  consider- 
able eminence,  concerning  the  principles  of  the  Differential  Calculus,  of 
which  Rolle  was  a  strenuous  opposer. 


FIGUKATE  AND  POLYGONAL  NUMBERS.  345 


Ex.  7.  It  is  required  to  find  the  value  of  ^ ^,  when 

X  is  equal  to  a.  Ans.  (2a)   . 

1 -p» 

Ex.  8.  It  is  required  to  find  the  value  of  the  expression , 

1  — X 

when  X  is  equal  to  1 .  Ans.  n. 

Ex.  9.  It  is  required  to  find  the  value  of  the   expression 

a  '\/ doc ^'^x 

—-^ =:r  ,  when  X  is  equal  to  a,  Ans.  8a. 

a—  ^ax 
Ex.  10.  It  is  required  to  find  the  value   of  the  expression 

nx"^^-{n-[-\)x"+\       ,  .  ,    — 
^ — ^ ,  when  X  is  equal  to  1. 


Ans. 


n{n+\) 


Ex.  11.  Tt  is  required  to  find  the  value  of  the  expression 

'\/x—-y/a-\r-\/{^  —  (^)      I,  •  1 .  A  1 

yr— — -^^ -y  when  X  is  equal  to  a.  Ans.  — :=.. 

V'(a:2-a2)  ^  ^2a 

^  III.    ON  FIGURATE    AND    POLYGONAL    NUMBERS. 

522.  Figurate  Numbers,  are  such  as  arise  from  taking 
the  successive  sums  of  the  series  of  natural  numbers  1,  2,  3, 
4,  5,  &c. ;  and  then  the  successive  sums  of  these  last,  and  so 
on  :  and  polygonal  numbers,  are  those  which  are  formed  of 
the  successive  sums  of  the  terms  of  any  arithmetical  pro- 
gression beginning  with  unity  ;  each  of  ihem  being  usually 
divided  into  orders,  according  to  the  scale  of  their  generation, 
which,  as  far  as  regards  those  of  the  first  class,  may  be  shown 
as  foHows  : 

Order.  Fisurate  Nurrvbers.  Gen.  Terms. 

1 


Figurate  Nurrvbers. 
1,  2,  %  4,  5,  6,  &c. 


1,  3,  6,  10,  15,  21,  &c. 
1,  4,  10,  20,  35,  56,  &c. 


4  1,  5,  15,  35,  70,  126,  vtc. 

&c.  &c. 

Where  it  is  to  be  observed  that  the  general  terms,  here  given, 
are  so  called,  because  if  1,  2,  3,  4,  &c.  be  respectively  sub- 


n 
n(n+l) 

TTT2- 

n(n+l)(n  +  2) 

12     3 

;^(/^+l)(n  +  2)(/^  +  3) 
1.2.3     4 
&c. 


346 


FIGURATE  AND 


stituted  in  each  of  them,  for  n,  we  shall  obtain  the  several  terms 
of  the  series. 

And  if,  instead  of  the  natural  numbers  1,2,  3,  4,  &c.  which 
give  triangular  numbers,  an  arithmetical  series  be  taken,  the 
common  difference  of  which  is  2,  the  sum  of  its  successive 
terms  will  be  the  series  of  square  numbers  ;  if  the  common 
difference  be  3,  the  series  will  be  pentagonal  numbers  ;  if  4, 
hexagonal ;  and  so  on  :  thus, 
Arith.  Series. 
1,  2,  3,  4,  &LC. 


3,  5,  7,  &c. 
10,  &c. 
1,  5,  9,  13,  &c. 


1,  4,  7, 


Ord. 
1 


Polygonal  Numbers. 

Gen.  Terms. 

Tri.  1,  3,  6,  10,  &c. 

«(n-f  1) 

Sqrs.  1,  4,  9,  16,  &Lc 

n(2n-f0) 
2 

Pent.  1,  5,  12,22,  &c. 

n{Zn  —  l) 

9 

Hex.  1,  6,  15,  28,  &c. 

n(4n-2) 
2 

&c. 

&c. 

&c. 

Where  the  number  denoting  any  order,  is  the  common  differ- 
ence  of  the  arithmetical  series,  from  which  the  polygonal  num- 
bers, belonging  to  that  order,  are  generated. 

In  like  manner,  if  w^e  take  the  successive  sums  of  the  se- 
veral polygonals  thus  obtained,  and  then  the  successive  sums  of 
these  last,  and  so  on,  a  great  variety  of  other  orders  of  series 
of  this  kind  may  be  readily  obtained. 

Hence,  also,  in  general,  if  n  be  made  to  denote  the  number 
of  terms  of  the  series,  a  figurate  of  any  m,  may  be  expressed 
by  the  following  formula. 

n     n-fl      n  +  2                                              n4-(m  — 1) 
1^     2     ^~3~ m         • 

And  supposing  n  to  be  the  number  of  terms  of  the  series, 

as  before,  a  polygonal  number  of  the  order  m  — 2,  or  one  that 

has  the  number  of  its  sides  denoted  by  w?,  may  be  expressed 

{m—2)n'^~{m'-^)n 
by w-^ . 

So  that  figurate  numbers,  of  any  order,  may  be  always  de- 
termined, without  computing  those  of  the  preceding  orders, 
by  taking  as  many  factors,  in  the  first  of  these  formulae,  by 
•substituting  the  number  denoting  that  order  for»j— 2,  or  the 
number  of  sides  of  the  polygon,  for  w,  and  taking  n  equal  to 
the  term  required. 

Example  1.  Required  the  15th  term  of  the  second  order  of 
figurate  numbers,  1,  3,  6,  10,  15,  &c. 


POLYGONAL  NUMBERS.  347 

Here  m  being  =2,  and  n  =  15,  we  shall  have  hy  the  first 
formula, 

!^l)=l£iH±i)=H2il^  =15  X8  =120. 

<i  ^  li  • 

the  term  required. 

Ex.  2.  It  is  required  to  find  the  12th  term  of  the  fifth  order 
of  polygonal  numbers,  being  those  called  heptagonal,  or  such 
as  would  be  represented  by  a  figure  of  seven  sides. 

Here  m  being  equal  7,  and  n  =  12,  we  shall  have,  by  the 
second  formula, 
(m-2K-(m-4)n^(7-2)xl44-(7-4)Xl2^^^^^^     ^ 

■^  >o 

X6  =  360  — 18  =  342,  the  term  required. 

Ex.  3.  It  is  required  to  find  the  20th  term  of  the  5th  order 
of  figurate  numbers.  Ans.  42504 

Ex.  4.  It  is  required  to  find  the  1 3th  term  of  the  9th  order 
of  figurate  numbers.  Ans.  293930. 

Ex.  5.  It  is  required  to  find  the  36th  term  of  that  order  of 
polygonal  numbers,  which  is  denoted  by  a  figure  of  twenty- 
five  sides.  Ans.  14526 


CHAPTER  XVII. 

ON 

INDETERMINATE  AND  DIOPHANTINE 
ANALYSIS. 

^  I.    ON  INDETER-AIINATE  ANALYSIS 

523.  When  the  enunciation  of  a  question  does  not  furnish 
as  many  equations  as  there  are  unknown  quantities  to  be  de- 
termined, the  question  is  said  to  be  indeterminate,  being  usually 
such  as  admit  of  a  great  variety  of  solutions  ;  although,  when 
the  answers  are  required  only  in  whole  positive  numbers,  they 
are  generally  confined  within  certain  limits  :  the  determina- 
tion of  which  forms  a  particular  branch  of  Algebra,  called 
Indeterminate  Analysis, 


348  INDETERMINATE   ANALYSIS. 

To  begin  with  one  of  the  easiest  questions  ;  let  there  be 
required  two  positive  integer  numbers,  the  sum  of  which  is 
equal  to  10. 

Let  us- represent  them  by  x  and  y  ;  then  we  have,  or+y 
=  10,  and  a;=10 — y,  where  y  is  so  far  only  determined  that 
it  must  represent  an  integer  and  positive  number.  We 
may  therefore  substitute  for  it  all  integer  and  positive  num- 
bers from  1  to  infinity  ;  but  since  x  must  likewise  be  a  posi- 
tive number,  it  follows,  that  y  cannot  be  greater  than  10  ;  be- 
cause X  must  be  positive  ;  and  if  we  also  reject  the  value  a;  =  0, 
we  cannot  make  y  greater  than  9 ;  so  that  only  the  following 
solutions  can  take  place  : 

If  y  =  l,2,  3,  4,  5,  6,  7,  8,  9, 
then  x:rr9,  8,  7,  6,  5,  4,  3,  2,  1. 

But  the  four  last  of  these  nine  solutions  being  the  same 
as  the  four  fir.st,  it  is  evident,  that  the  question  really  admits 
only  of  five  different  soiutions. 

524.  As  we  have  found  no  difficulty  in  this  question,  we 
may  proceed  to  others,  which  require  different  considera- 
tions. 

Problem  1.  To  find  the  values  of  the  unknown  quantities 
X  and  y  in  the  equation 

aa:4;5yr=c,  or  ax-{-by=c, 
where  a  and  b  are  given  numbers  which  admit  of  no  common 
divisor,  except  when  it  is,  also,  a  divisor  of  c. 

RULE. 

525.  I.  Let  wh.  denote  a  whole  or  integral  number,  and 
reduce  the  equation  to  the  form  3?=::; -^^^ z=wh. 

2.  Make  — =  -^ ,  by  throwing  all  whole  numbers 

out  of  it,  till  d  and  e  be  each  less  than  a. 

3.  Find  the  difference,  or  sum,  of  -~ ,  or  some  mul- 

a 

OLV 

tiple  of  it,  and  — ,  or  any  other  multiple  of  it  that  comes  near 

the  former,  and  the  result  will  be  a  whole  number. 

4.  Take  this,  or  anymuliiple  of  it,  from  one  of  the  fore- 
going fractions,  or  from  any  whole  number  which  is  nearly 
equal  to  it,  and  the  result,  in  this  case,  will  also  be  a  whole 
number. 


INDETERMINATE  ANALYSIS.  349 

5.  Proceed  in  the  same  manner  with  this  last  result,  and 
80  on,  till  the  coefficient  of  y  becomes  equal  to  1,  or 

a  ^ 

6.  Then  will  y^ap^^r^  where  p  may  be  any  whole  num- 
ber whatever,  that  makes  y  positive;  and  as  the  value  of 
y  is  now  known,  that  of  a:  may  be  found  from  the  given  equa- 
tion. 

Example  1.  Given  2a:+3y=25,  to  determine  x  and  y  in 
whole  positive  numbers.  • 

Herex=Hi^?-?^=12-y+i^. 

Hence,  since  x  must  be  a  whole  number,  it  follows  that 

must  also  be  a  whole  number. 

2 

Let,  therefore,        ^ =wh.z=p  ; 

then  1  —y—2p,  or  y  =  1  — 2/>. 
And  since  

a:=:12-y-h^^=12-(l-2j9)-f;)  =  12+3;i-l, 

we  shall  have  jr^i^ll-fSp,  andy=l— 2p  ; 
where  p  may  be  any  whole  number  whatever,  that  will  render 
the  values  of  x  and  y  in  these  two  equations  positive. 

But  it  is  evident,  from  the  value  of  y,  that  p  must  be  either 
0  or  negative,  and,  consequently,  from  that  of  x,  that  it  must  be 
—  1,-2,  or — 3. 

Whence, j9  =  0,/)  —  —  1,J3  =  — 2,p= — 3  ; 

,  i  Xz=\\,  X=zS,  X-=:zd.  Xz=z2  \ 

then  <  1  o  c         ^ 

\y—   l,y=3,y  =  5,y  =  7; 

which  are  all  the  answers  in  whole  positive  numbers  that  the 

question  admits  of. 

Ex.  2.  Given  21a;-4-17y=r2000,  to  find  all  the  possible  va- 
lues of  X  and  y  in  whole  numbers. 

Here  x=i ^=n95H ---^  —  wh.\ 

or  omitting  the  95,  — —~-=ich. ; 
Zl 

1      u        i:,-  .        21y  .  5-17y     4y+5        , 
consequently,  by  addition,  -^,-H oT  21 — ^^^'^^ 

*i       "^y-^^     «     20y4-25      ,   ,  4  +  20y        , 
Also.  JL__x5  =  -J^.=  l+-^=.A.; 

31 


92 

75 

58 

41 

24 

4 

25 

46 

67 

88 

350  INDETERMINATE  ANALYSIS. 

or,  by  rejecting  the  whole  number  1, -~zzz.wh. 

.    ^  »  y.         .        21y     4  +  20y     w— 4 

And,  by  subtraction,  —f —-^—^———wh.^-p  ; 

Z\.  21  21 

whence  i/  =  21p-{-4, 

.         2000-17y     2000-17(21^+4)     ^„     ,^ 

Whence,  if  p  be  put  =  0,  we  shall  have  the  least  value  of  y  =:  4, 
and  the  corresponding,  or  greatest,  value,  of  a;  —92. 

And  the  rest  of  the  answers  will  be  found  by  adding  21 
continually  to  the  least  value  of  y,  and  subtracting  17  from  the 
greatest  value  of  x ;  which  being  done,  we  shall  obtain  the 
six  following  results  : 

7 

109 
These  being  all  the  solutions  the  question  admits  of. 

Ex.  3.  Given    19a;  =  14y— 11,  to  find  x  and  y  in   whole 

numbers. 

_j  14y-ll         J  ,  19y        . 

Here  x= — ^— — =zwh.,  and  -~-=vm. ; 

V            u        t,         •        19y       14y-ll      5y+ll        , 
whence,  by  subtraction,  — -^- ^-— — =         — z=wn. 

5y-f  11  20y-|-44  „  ,  y  +  6 

Also,  J^X4=-^-=y+2+^=»A.; 

and  by  rejecting  y+2,  which  is  a  whole  number, 

^        =wh.=p\  .-.^==19^— 6,  and 

_14y--ll_14(19;?-6)-ll_266jP-95_^^ 

''-'    19    -       r9       -    19    -i4p-5. 

Whence,  if  p  be  taken  =1,  we  shall  have  a?  =  9  and  y=13, 
for  their  least  values  ;  the  number  of  solutions  being  obviously 
indefinite. 

526.  When  there  are  three  or  more  unknown  quantities, 
and  only  one  equation  by  which  they  can  be  determined,  it 
will  be  proper  first  to  find  the  limit  of  that  quantity  which  has 
the  greatest  coefficient,  and  then  to  ascertain  the  diff'erent  va- 
lues of  the  rest,  by  separate  substitutions  of  the  several  values 
of  the  former,  from  1  up  to  the  extent  required,  as  in  the  fol- 
lowing example. 

Ex.  4.  Given  3x  +  5i/-^7z  =  \00,  to  find  all  the  different 
values  of  x,  y,  and  z,  in  whole  numbers. 

Here  each  of  the  least  integer  values  of  x  and  y  are  1,  by 
the  question  ;  whence  it  follows,  that 


INDETERMINATE  ANALYSIS.  351 

100-5-3     100  —  8     92      ,„, 

Consequently  z  cannot  be  greater  than  13,  which  is  also 
the  limit  of  the  number  of  answers  ;  though  they  may  be  con- 
siderably less. 

By  proceeding,  therefore,  as  \i\  the  former  rule,  we  shall 
have 

100— 5y-7;2     ^^  ^     ,  l-2y-^ 

X  — -^ r=:33— y— 2^H f r=to/<.; 

and  by  rejecting  33— y  — 2^, 

\—2y—z         ,  ^y      \—1y—z     y-\-\~z 

— 3 — =^^-'  "^Y+— T— =^^-3— =«'^-=^- 

Whence,  y  =  3j9  +  ^  — 1  ;  and,  putting  J9  =  0,  we  shall  hav« 
the  least  value  of  y=-z  —  1  ;  where  z  may  be  any  number 
from  1  up  to  1 3,  that  will  answer  the  conditions  of  the  ques 
tion. 

When,  therefore,  2^=1,  we  have  y=0, 

100-7     ^, 
and  a;  =  — - — =31- 

And  by  taking  ^  =  2,  3,  4,  5,  <fcc.  the  corresponding  values 
of  X  and  y,  together  with  those  of  z^  will  be  found  to  be  as 
below. 

8 
7 
3 

Which  are  all  the  integer  values  of  a:,  y,  and  z,  that  can  be 
obtained  from  the  given  equation. 

527.  If  there  be  three  unknown  quantities,  and  only  two 
equations,  exterminate  one  of  these  quantities  in  the  usual 
wa}',  and  find  the  values  of  the  other  two  from  the  resulting 
equation,  as  before  ;  then,  if  the  values,  thus  found,  be  sepa- 
rately substituted,  in  either  of  the  given  equations,  the  cor- 
responding values  of  the  remaining  quantities  will  likewise 
be  determined, 

Ex.  5.  Given  a'~2y-f  ;^=5,  and  2j:+y— .3=7,  to  find  the 
values  of  a:,  y,  and  z. 

Here,  by  multiplying  the  first  of  these  equations  by  2,  and 

subtracting  the  second  from  the  product,  we  shall  have 

34-5y  2y 

3^-5y=3,  or  ;^=,_L_^:3=I-fyH-|-=toA.  ; 

and  consequently  -J^,  or  — ^ ^=^z=wh.z=zp 

whence  i/=Sp. 


z  =  l 

0 

3 

4 

5 

6 

7 

y  =  0 

1 

2 

3 

4 

5 

6 

a;=:31 

27 

23 

19 

15 

11 

7 

5 

6 

7 

8 

9 

3 

6 

9 

12 

15 

6 

11 

16 

21 

26 

353  INDETERMINATE  ANALYSIS. 

And  by  taking  p=0,  1,  2,  3,  4,  &c.  we  shall  have  y=0, 
3,  6,  9,  12,  15,  &c.  and  z=],  6,  11,  16,  21,  &c. 
But  from  the  first  of  the  two  given  equations 
a;=5-f  2y — z  ; 
whence,  by  substituting  the  above  values  for  y  and  z,  the  re- 
sults will  give 

XZZZ4,  5,  6,  7,  8,  9,  &c. 
And  therefore  the   first  six  values  of  x,  y,  and  z^  are  as 
below : 

ar=4 
y=:0 
z=l 

Where  the  law  by  which  they  can  be  continued  is  suffi- 
ciently obvious. 

Ex.  6.  Given  3a:  =  8y  — 16,  to  find  the  least    values  of  ar, 

and  in  whole  numbers.  Ans.  x=8,  i/  =  5. 

Ex.  7.   Given  14a?=5y+7,to  find  the  least  values  of  x  and 

y  in  whole  numbers.  Ans.  a;=3,  y=7. 

Ex.  8.  It  is  required  to  divide  100  into  two  such  parts,  that 

one  of  them  may  be  divisible  by  7,  and  the  other  by  11. 

Ans.  The  only  parts  are  56  and  44. 
Ex.  9.  Given  lla?+5y=254,  to  find  all  the  possible  values 
of  X  and  y  in  whole  numbers. 

.        (0^=19,14,9,4, 
^"^- >  y=:9,  20,  31,42. 
Ex.  10.  Given  17x-f-19y+2l2r.T=400,  to  find  all  the   an- 
swers in  whole  numbers  which  the  question  admits  of. 

Ans.  10  different  answers. 
Ex.   11.  Given  5ar-f-7y-h  11^^=224,  to  find  all  the  possible 
values  of  a:,  y,  and  z,  in  whole  positive  numbers. 

Ans.  The  number  of  answers  is  59. 
Ex.   12.  A  person  bought  as  many  ducks  and  geese,  to- 
gether, as  cost  him  28^.  ;  for  the  geese  he  paid  4s.  Ad.  a  piece, 
and  for  the   ducks  2s.  6d.  a  piece  ;  what  number  had   he  of 
each?  Ans.  3  geese  and  6  ducks. 

Ex.  13.  How  many  gallons  of  spirits,  at  12^.,  15^.,  and 
18 jr.  a  gallon,  must  a  rectifier  of  compounds  take  to  make  a 
mixture  of  1000  gallons,  that  shall  be  worth  17  shillings  a, 
gallon?         Ans.  lllj  at  12.?.,  111^  at  15^.,  and  777J  at  18;?. 

PROBLEM. 

528.  To  find  such  a  whole  number,  as,  being  divided  by 
other  given  numbers,  shall  leave  given  remainders 


INDETERMINATE  ANALYSIS.  353 


1.  Call  the  number  to  be  determined  a:,  the  numbers  by 
which  it  is  to  be  divided  a,  &,  c,  &c.,  and  the  given  remain- 
ders/, g,  h,  &LC. 

2.  Subtract  each  of  the  remainders  from  a?,  and  divide  the 

differences  by  a  ;  and  there  will  arise ^, —^  ' ,  6lc 

a  a  a 

=  whole  numbers. 

x—f 

3.  Put  the  first  of  these  fractions  — —=p,  and  substitute 

a 

the  value  of  x,  as  found  from  this  equation,  in  the  place  of  x 
in  the  second  fraction. 

4.  Find  the  least  value  o{ p  in  this  second  fraction,  by. the 
last  problem,  which  put  =zr,  in  the  place  x  in  the  third  frac- 
tion. 

529.  Find,  in  like  manner,  the  least  value  of  r,  in  this  third 
fraction,  which  put  =s,  and  substitute  the  value  of  a;,  in  terms 
of  s,  in  the  fourth  fraction,  as  before  ;  and  so  on,  to  the  last ; 
when  the  value  of  x  thus  found,* will  give  the  whole  number 
required. 

Example  1.  It  is  required  to  find  the  least  whole  number, 
which,  being  divided  by  17,  shall  leave  a  remainder  of  7,  and 
when  divided  by  26,  shall  leave  a  remainder  of  13. 

Let  x=z  the  number  required. 

Then and  — — -=  whole  numbers. 

jp 7 

And,  putting  =p,  we  shall  have  x=llp-\-l  ;  which 


value  of  a:,  being  substituted  in   the   second  fraction,  gives 
wh'. 


17;? 4-7-13  _17p-6 


26  26 

„      .    .      ^  .         ,         6o       17p  — 6     9»4-6 

But  It  IS  obvious  that  -—- ^— — =-%^ — =^wh.  ; 

9p^&     ^     27/;+18         ,/?+18        , 

And  by  rejecting  p,  there  remains  --— — =t/?A.=r ; 

therefore  p=26r— 18 ; 
where,  if  r  be  taken  =1,  we  shall  liave  p=iQ. 

And  consequently  a-— 17;?-^7=17x8  +  7=143,  the  num- 
ber required.  31* 


354  INDETERMINATE  ANALYSIS. 

Ex.  2.  To  find  a  number,  which,  being  divided  by  6,  shall 
leave  the  remainder  2,  and  virhen  divided  by  13,  shall  leave 
the  remainder  3.  Ans.  68. 

Ex.  3.  It  is  required  to  find  the  least  whole  number,  which, 
being  divided  by  39,  shall  leave  the  remainder  16,  and  when 
divided  by  56,  the  remainder  shall  be  27.  Ans.  1147. 

Ex.  4.  It  is  required  to  find  the  least  whole  number, 
which,  being  divided  by  11,  19,  and  29,  shall  leave  the  re- 
mainders, 3,  5,  and  10.  Ans.  4128. 

Ex.  5.  It  is  required  to  find  the  least  whole  number, 
which,  being  divided  by  each  of  the  nine  digits,  1,  2,  3,  4,  5, 
6,  7,  8,  9,  shall  leave  no  remainder.  Ans.  2520. 

PROBLEM. 

On  Compound  Indeterminate  Equations. 

530.  Equations  of  this  kind,  not  higher  than  the  second 
degree,  which  admit  of  answers  in  whole  numbers,  are  chiefly 
such  as  consist  of  the  products,  or  squares,  of  two  unknown 
quantities,  together  with  the  quantities  themselves  ;  being, 
usually,  one  of  the  four  general  forms  given  in  the  following 
rule. 

RULE. 

1.  If  the  equation  be  of  the  form  cvi/=ax-\-bi/-^c,  we  shall 
have,  for  its  solution  in  whole  numbers,  y=a-\ ;  where 

X  —  0 

X — b  must  be  a  divisor  of  ab-^c. 

2.  If  the  equation  be  of  the  form  x'^-\-xi/  =  ax-\-bi/-\-c,  we 

shall  have 

c+b{a^b)  ^ 
y=_a;+a_J-t— — -^ ; 

where  x — b  must  be  a  divisor  o(  c-\-b{a  —  b). 

3.  If  the  equation  be  x'^=y^-\-ay-j-by  we  shall  have  y— 
^2 — 4j      j^ — a  a  ,  ,  ,  , 

— f-  ,  and  x=--\-y  —  n  ;  where  a  and  n  must  be  even 

numbers,  and   n  be  so   taken   that    8w  may  be   a  divisor  of 
a^  —  Ab. 

4.  If  the  equation  be  a;2=ay2-f5y+c2,  we  shall  have  y  = 

— ,  and  a:=c4-ny  ;  where   n  must  be  some  whole  num- 

n^ — a  # 

ber  between  -y/a  and  — -. 
^  2c 


INDETERMINATE  ANALYSIS.  350 

Example  1.  Given  a;y =42— 2a;— 3y,  to  find  tlie  several  va- 
lues of  X  and  y  in  whole  numbers. 
Here,  by  the  first  form, 

a— —2,  b=—3,  and  c==42, 

o  .  6+42         „  ,      48 
whence  y=-2  +  -^3-=-2+-p3. 

Where  it  is  plain,  that  x  must  be  such  a  number,  that,  when 
added  to  3,  it  shall  be  a  divisor  of  48.  But  the  divisors  of 
48,  that  will  give  quotients  greater  than  2,  are  16,  12,  8,  6,  4, 
and  2. 

And  consequently  the  integral  values  of  the  two  unknown 
quantities  are 

a;=16  — 3,  or  13  |  =12—3,  or  9  |  =8—3,  or  5  | 
=6—3,  or  3  I  =4—3,  or  1. 


s-.- 

-l?-2 
12 

,or  2 

=?-..o.4 

48 
~"6  " 

-2,  or  6 

_48 
4  ■ 

-2,  or  10. 

Which  are  all  the  answers  in  whole  positive  numbers  that  the 
question  admits  of. 

Ex.  2.  Given  x^  =  ij'^-\-207/,  to  find  the  values  of  x  and  y  in 
whole  positive  numbers. 

Here,  by  the  third  form,  a =20,  and  b  =0, 

400  .  n— 20     50     n     ,  • 
whence,  y=~^ — I ^ —  = ho""!^'  ^^^  x=lO  +  i/—n. 

Where  it  is  plain,  that  n  must  be  some  even  number  which 
is  a  divisor  of  50. 

But  the  only  number  of  this  kind,  that  will  give  positive  re- 
sults, is  2. 

•.y=5^+l_10=16,  and  a:=10  +  16-2=24. 

Ex.  3.  Given  x^  =  5y^-~l2i/-\-64f  to  find  the  values  of  x  and 
y  in  whole  positive  numbers. 

Here,  by  the  4th  form,  a=5,  5  =  — 12,  and  c=8. 

-12-16n     16(n— f)        ,         ^  , 
Whence,  y= — - — =  — -^ r^,  and  x=8-\-ni/. 

^  TL^  —  5  5 TV" 

Where  it  is  plain,  that  n  must  be  less  than  the  -v/5,  and  great- 
er than  \ ;  which  numbers  are  only  1  and  2. 

—  12-32 
4—5 
and  :c=8+lx 7=15  |  =8+2x44=96. 


-12-16     ^ 
^         1-5 


=44. 


356  DIOPHANTINE  ANALYSIS. 

Ex.  4.  Given  x^-\-xi/=2x-\-3i/-\~29,  to  find  the  values  of  « 
and  t/  in  whole  positive  numbers. 

^^^•^y=21,7. 
Ex.  5.  It  is  required  to  find  two  numbers,  such,  that  their 
product,  added  to  their  sura,  shall  be  79. 

Ans.  ^3^^  19,  15,  19. 
Ex.  6.  Given  x'^+orr/=z4x+3f/-{-27f  to  find  the  several  va- 
lues of  X  and  y  in  whole  numbers. 

Ans  J^=  ^'    5,  and  6, 
^''^•<y=:27,  ll,and5. 
Ex.  7.  Given  x2rry2_^  j  oOyH- 1000,  to  find  the  two  last  va- 
lues of  X  and  y,  in  whole  numbers. 

Ans.  a: =70,  and  y=30. 
Ex.  8.  GivQp  a;2=50y2+i00y+100,  to  find  the  values  of  a? 
and  y  in  whole  numbers. 

Ans.  a:=190,  and  y=40. 

^  II.    ON  THE  DIOPHANTINE  ANALYSIS. 

531.  The  Diophantine  Analysis  relates  chiefly  to  the  find- 
ing of  square,  cube,  and  other  similar  numbers,  or  the  rendering 
certain  compound  expressions  free  from  surds  ;  the  principal 
methods  of  effecting  which  are  comprehended  in  the  following 
problems. 

PROBLEM  I. 

532.  To  render  surd  quantities  of  the  form  ^/(a-\-hx-\-cx'^) 
rational ;  or,  to  find  such  values  of  x  as  will  make  a-\-hx-{'Cx'^ 
a  square. 

Case  1.  When  the  expression  is  of  the  form  ^/{a-^hx),  that 
is,  when  cz=.0.  Put  ^/{a-\-hx)=^n,  or  a-\-hx-=n'^ ;  and  we  shall 

w^ — d 
have  0!= — 7 —  ;  where  n  may  be  any  number,  either  integral 

or  fractional,  that  will  render  the  value  of  x  positive. 

Example  1.  It  is  required  to  find  a  number,  such,  that  if  it 
be  multiplied  by  5,  and  then  added  to  19,  the  result  shall  be 
a  square. 

;>2_19 
Let  5a:4-l9=n2,  or  x= — - —  ; 
5 

where  n  may  be  any  number  whatever  greater  than  ^\  9. 
Whence,  if  n  be  taken  =5,  6, 7,  respectively,  we  shall  have 


DIOPHANTINE  ANALYSIS.  357 

25-1^     ,,        36-19     ^2        49-19     ^ 

the  latter  of  which  is  the  least  value  of  jc,  in  whole  numbers, 
that  will  answer  the  conditions  of  the  questioti ;  and  conse- 
quently 

5x4-19=5x6+19  =  304-19=49, 
a  square  number,  as  was  required. 

533.  Ex.  2.  Find  a  number  such,  that  if  it  be  multiplied 

by  5,  and  the  product  increased  by  2,  the  result  shall  be  a 

square. 

n^— 2 

•       Put  5x^-2=11^,  then  x  = ; 

5 

if  we  assume  n=2,  then  a;=f  ;  and  by  assuming  other  values 

for  «,  different  values  of  x  may  be  obtained. 

534.  Case  2.  When  the  expression  is  of  the  form  ■\/(hx-{' 
cx^)  ;  that  is,  when  a  —  0. 

Put  y^(bx-\-cx^)=:nx;  .'.  bx+cx'^z=n'^x'^^  then  b-\-cx=:n^x; 

whence  x=— ,  and  whatever  value  may  be  given  to  n  in 

this  expression,  there  will  result  a  value  of  x  that  will  make 
^^(bx-^-cx^)  rational. 

Example  1.  .It  is  required  to  find  an  integral  number,  such, 

that  it  shall  be  both  a  triangular  number  and  a  square. 

It  is  here  first  to  be  observed,  that  all  triangular  numbers 

X"  -\-  X 
are  of  the  form  — - —  ;  and  therefore  the  question  is  reduced 

to  the  making  — — — ,  or  it  equal  — a  square.    But  since 

a  square  number,  when  multiplied,  or  divided,  by  a  square 
number  is  still  a  square  ;  it  is  the  same  thing  as  if  it  were 
required  to  make  2x2-[-2a:  a  square.         • 

TTl^X^ 

Let  therefore  2x^-\-2x=z — ^,  then  dividing  by  x,  andmul- 

n^  " 

tiplying  the  result  by  n^,  the  equation  will  become  2vP'x-\-2r? 
^=.n^x  ;  and  consequently 

__     2n^ 

Where,  if  n  be  taken  =2  and  wi=3,  we  shall  have 

,  x'+x      64  +  8     72     „^ 
*==8,  and  --2L-=_L_=:     =36, 


358  DIOPHANTINE  ANALYSIS. 

for  the  least  integral  triangular  number  that  is  at  the  same 
time  a  square. 

535.  Ex.  2.  Find  a  number  such,  that  if  its  half  be  added 
to  double  its  square,  the  result  shall  be  a  square. 

Let  X  denote  the   number,  then  we  must  have  'Zjc^-^-^x^l  a 

square  ^^ri^x"^,  or  2x-\-\=n'^x  \  therefore,  a:  = — ,  n  be- 

2n2__4 

ing   any   number  whatever  :  if  n—2,    then  xz=: _.-,  a 

8 — 4     4 
square  number. 

536.  Case  3.  When  a  is  a  square  number,  put  it  equal  to 

<^2,  and  make   'y/(d'^-\-bx-\-cx'^)  =  d-\-nx\  i\i^.i\  d'^-\-dx-{-cx'^=z 

d'^-\'2dnx-\-n^x'^,ox  b^cx  =  2dn-\~n'^x  ;  and  consequently,  x=: 

2dn—b      ^     .. ,      ^  2dn 

-.     Or,  if  5  =  0,  x= -. 

c—n^  c—n?- 

Example  L  It  is  required  to  divide  a  given  square  number 
into  two  such  parts,  that  each  of  them  shall  be  a  square 
number. 

Let  c^— .  the  square  to  be  divided,  x'^'=.  one  of  its  square 
parts,  a^  —  x^zzz  the  other  ;  which  is  also  to  be  a  square. 

Put  a^ — x'^={nx—af':=n^x'^—2anx-\-a^i  and  we  shall  have 

2anx  =.  n^x^-^x"^,  or  n'^x-\-x=^2an  ;    and    consequently   x= 

2an                              2an  2an^        an'^-^a      an^—a 

— „-r--.  and  nx—a=  „  .  . a= 


Hence,  (-y— j^and  (     21?)^   ^^®   ^^*®    parts   required; 

where  a  and  n  may  be  any  numbers  whatever,  provided  n  be 
greater  than  unity. 

537.  Ex.  2.  Find  two  numbers,  whose  sum  shall  be  16, 
and  such,  that  the  sum  of  their  squares  shall  be  a  square. 

Let  x=:  one  of  the  numbers,  then  16— a;  denotes  the  other, 
and  we  have  to  m^e  x'^+{x  —  l6)^,  or  2a.2— 32a?+256,  a 
square. 

Put  2x'^  —  32x+256  =  {nx—l6y  =  n^x'^  —  32nx-j-256  ; 

hence,  2x'^—32x=n^x^  —  32nx,  and  2a;— 3'2=:»2a;— 32n  ; 

.    ,  32(71—1) 

consequently  x=  — ^ — -— . 
n^—2 

If  we  take  n  =  3,  we  shall  have  a'5=9^  ;  therefore  the  two 

numbers  are  9i  and  6^. 

538.  Case  4.  When  c  is  a  square  number,  put  it  =6^,  and 
'\/{a+bx-\-e'^x'^)z=n-\-ex  ;   then,  a-{-bx-\-e^x^  =  n^  +  2cna?4- 

e^x^,  or 


DIOPHANTINE  ANALYSIS.  359 


a — n^ 
2en — b' 


Or,i{b=0,x:    '""''' 


2en 

Example  1.  It  is  required  to  find  the  least  integral  number 
such,  that  if  4  times  its  square  be  added  to  29,  the  result 
shall  be  a  square. 

This  being  the  same  thing  as  to  make  4a:2-f  29  a  square  • 
let  4a;2_|.29=(2x+n)2=4a:2  +  4na  +  n2.     Then,  4na:+n2=29; 

or4na:=29-n*'';  .-.  a?=— ^^  ;  where,  if  n  be  taken  equal 

to  1,  we  shall  have  a?=— ^=^  =  7,  which  is  the  only  in- 
tegral number  that  answers  the  conditions  of  the  question. 

539.  Ex.  2.  Find  a  number  such,  that  if  it  be  increased  by 
2  and  5  separately,  the  product  of  the  sums  shall  be  a  square. 

Let  x=  the  number,  then  we  have  to  make  (a:+2)  (a:+5), 
or  a:2+7a;-fl0,  a  square,  which  denote  by  lx—nY\  then! 
a;2+7rr4-10  =  a:2-2«a'+w2,  or 

^2— 10 


7a:+10=— 2na:+n 


Xrr- 


7  +  2n' 

If  we  take  n=4,  we  shall  have  x=z-. 

5 

540.  Case  5.  When  neither  a  nor  c  are  square  numbers, 
yet  if  the  formula  can  be  resolved  into  two  simple  factors, 
(which  it  always  can  when  b^  —  4c  is  a  square,  but  not  other- 
wise), the  irrationality  of  it  may  be  taken  away,  by  putting 

V{a+bx-^cx-^)=  ^\(d-^ex)  {f^  gx)\=.n{d+ex)  ;  in    which 
case  we  shall  have 

{d-{-ex){f+gx)=:n^(d-\-ex)\  or  f+gxz=n^{d+ex) ; 

J  ,  dn^-f 

and  consequently  x= -. 

g — en^ 

Or,  if  (f=0,  x=  -4—,  and  if /=:0,  x-.       ^'^^ 


en^—g  -"  g-en^' 

The  two  factors  above  mentioned  will  be  found  by  putting 
c+iaj4-ca?2=0  ;  and  solving  this  equation,  we  shall  have 

'=-i+yyF^-4ac),  and  a:=-A--i  V(62-4ac); 
or  putting  ^(b'^—Aac)=5^,  the  values  of  a;  are 


360  DIOPHANATINE  ANALYSIS. 

and,  consequently,  (cx-\ — - — ),  and  (x-\ — - — ),  are  the  factors 
required. 

Example.  It  is  required  to  find  such  a  value  of  a?,  that 
•\/(6+13a:+6x2)  shall  be  rational,  and  consequently  6+ 13a; 
4-6a;2  a  square. 

Let  6a:2+13a;-f-6=0; 
and  solving  this  equation,  we  shall  have  x=  — f ,  and  x=z  —J  : 
therefore  the  two  factors  are  2x-f-3,  and  3a?4-2. 

TTI?  7n^ 

Put  (2a:4-3)(3a;+2)=— (3a:4.2)2,  or  2a:+3=— (3a;+2), 
n  n 

and  consequently,  by  reduction,  a;-=— — ^ — — j. 

Where  it  appears,  that,  in  order  to  obtain  a  rational  answer, 


m 


—  must  be  less  than  f ,  and  greater  than  f . 

Whence,  if  7n  =  6,  and  n=5,  we  shall  have 

3X25-2X36      75-72-       3     ,         .  .     , 

a;= =7777^ 7:=—;^^  the  value  required. 

3X36-2X25     108—50     58'  ^ 

Case  6.  When  neither  of  the  foregoing  will  apply,  if  the 
formula  can  be  resolved  into  two  parts,  one  of  which  is  a 
square,  and  the  other  the  product  of  any  two  simple  factors. 

Put  '^(aA-hx-\-cx'^)  ^  'y/\{d+exY  ^  (f-^gx){h+kx)\  = 
(d-{-ex)-\-n(f-\-gx)  ;  in  which  case  we  shall  have  {d-\-exY-\r 
(f+gx){h+kx)  =  {d+exf+2n{d-{-ex){f^gx)  +  n\f^gx)\ 
or  h'{-kx=2n(d+ex)-\-n\f-\-gx) ; 

and  consequently,  xz=.  -^ — tJ—t — r- 
k—n{2e-tgn) 

Or,  if  d=0,  x=z^-—-- r. 

k  —  n{2e-{gn) 

Or,  if  the  part  in  this  case,  which  is  found  to  be  a  square, 
be  a  known  quantity,  put  ■y/{a-\-hx-]rCx^)  =  V\(^'^(^'^f^) 
(g-\-hx)=d-\-n{e-\-fx)\  ;  then  we  shall  have 

d;^^(e^fx){g+hx)=d'^-ir2dn{e-{-fx)-\-(e-^fxf 
or  g-\-hxz=2dn-\-{e-{-fx), 
and  consequently,  by  transposing  and  uniting  the  different 

e-\-2dn—g 
terms,  a;= — r — -^ — . 
A-/ 

Example  1.  It  is  required  to  find  such  a  value  of  a:,  that 
V(13a;2-f.  15a?+7)  shall  be  rational,  or  13d:2+15a;+7  a  square. 


DIOPHANTINE  ANALYSIS.        *        361 

Let  this  formula  be  separated  into  the  two  parts  (1  —xf  and 
6+17a;+12a:2. 

Then,  since  1 72—4(6  X 12),  which  is  equal  to  1,  is  a  square, 
the  latter  part  may  be  divided  in  the  %tors  3a; +2,  and  Ax-\- 
3  ;  and  consequently  the  original  fprmula  maybe  represented 

^  (l-x)2  +  (2  +  3a:)x(3  +  4a:). 

Hence,  putting  V(13a;24-15a:+7)  = 
y^j(l_a:)2-|-(2  +  3a:)x(34-4^)i=(l-a^)  +  «(2  +  3a:), 
we  shall  have  (1  -xY+{2-\-'ix)  X  (3  +  4)=(l  -xf+2n{\  -a:) 
X(2  +  3)  +  7i2(2  +  3a:)2,  or 

3  +  4a'  =  2n(l-a:)  +  n2(2  +  3a:)  ; 
^      .  2n4-2n2— 3 

and  consequently,  by  reduction,  ^^  Ay^n—^n^' 

2+2-3     1 
Where,  taking  n=l,  we  have  a?=^       _g  =  g  ; 

13     15  13  .  45  .  63     121 

and  13a;2+15a:+7  =  — 4-^-f7=-^4-9-+-9-=^- 

a  square  number,  as  required. 

Ex.  2.  Find  a  value  of  x,  such,  that  2j:2  +  8a;+7  shall  be 
a  square. 

This  expression,  after  a  few  trials,  is  found  to  be  equiva- 
lent to  (a;  +  2)2+(a:+l)x(a;  +  3),  which  being  equated  with 
j(^_f_2)-4t+l)|2  =  (a:2_|-2)2-2n(x  +  2)  X  (x+l)  +  n2(x+ 

1)2,  there  results 

xj-^  =  -2n{x-ir'^)'\-n\x-ir\)  \ 

n2  —  4rt  —  3 

whence,  a:r=  .   ,  ^ 5« 

i-f2n  — w2 

If  we  take  n=3,  we  shall  have  a:=3,  and  2x'^-\'Sx-\rl— 
49,  a  square,  as  was  required. 


PROBLEM    II. 


541.  To  render  surd  quantities  of  the  form  ^{a-]rhx-\-cx'^ 
-\-dx^)  rational,  or  to  find  such  values  of  x  as  will  make  a-^hx 
-fca:2-fda;3  a  square. 

This  problem  is  much  more  limited,  and  difficult  to  be  re- 
solved, than  the  former,  there  being  but  a  few  cases  of  it  that 
admit  of  answers  in  rational  numbers.  The  rules  for  obtain- 
ing them  are  of  such  a  confined  nature,  that  when  the  un- 
known quantity  has  more  than  one  value— which,  however, 
is  not  often  the  case— the  rest  can  onlv  be  determined  one  at 
32 


362  DIOPHANTINE  ANALYSIS. 

a  time,  by  repealing  the  operation  with  the  value  last  obtain- 
ed, as  often  as  may  be  found  necessary. 

«  RULE. 

542.  Case  1.  When  a=^,  and  5=0,  put  the  remaining 
part  '^(cx'^'\-dx^)  =  nx,  or  cx^-\-dx^  =  n'^x^ ;  then  we  shall  have 

71— -C 

c-{-dx=n^  ;   .'.x  =  — - — . 
a 

Where  n  may  be  any  number  whatever  greater  than  the  square 
root  of  c. 

Example  1.  It  is  required  to  find  such  a  value  of  a?  that 
y(3x24-lla:^)  shall  be  rational,  and  consequently  Saj^+H*' 
a  square. 

Let  ■x/{3x'^-{-\lx^)  =  nx,  or  3x^+11  x^  =  n'^x^. 

Then,  by  dividing,  we  shall  have  3-j-  IIx^ti^. 

fl2 3 

And  consequently  x=—- —  ;  where  n  may  be  any  number, 

positive  or  negative,  that  is  greater  than  -y/S. 

Taking  therefore,  n=2,  3,  4,  5,  &c.  respectively,  we  shall 

1      6     13    22 
have  ^=T-r>  tt,  tji  ttj  or  2,  the  last  of  which  is  the  least  in- 
tegral answer  which  the  question  admits  of. 

Ex.  2.  Find  a  number  such,  that  if  three  times  its  cube  be 
added  to  twice  its  square,  the  sum  shall  be  a  square. 
Here  we  must  make  3x^-\-2x'^  a  square  ; 

let  n'^x^  be  the  square,  then  3a:-|-2  =  n-; 


2 
If  we  take  n=3,  we  have  x=:3,  the  number  required. 
543.  Case  2.  When  a  is  a  square  number,  put  it  equal  to  e^, 

and  make  ■\/(e'^-\-bx-\-cx^-\-dx^)z=€-{-—-Xf  or  e^-{-bx  -\-cx'^-\- 

b  b^ 

dx^={e+---)x  =  e'^-{'bx+  -t-%^' 

Hence,  cx^-\-dx?=-—;rx.  and  by  division    and   reduction 

Note.  The  assumed  root  e-\-—x  is  determined  by  first  tak- 
ing  it  in  the  form  c+nar,  and  then  equating  the  second  term 


DIOPHANTINE  ANALYSIS.  363 

of  it,  when  squared  with  the  second  term  of  the  original  for- 

b 
inula;  in  which  case  n  will  be  found^:^^' 

Example  1.  It  is  required  to  find  such  a  value  of  x,  that 
l_|_2a;--j;2-f  a;3  shall  be  a  square. 

Here,  1  being  a  square,  let  i+2x-x'^'^x^  =  (l+xf=l-\' 
2x+x'^ ;  then,  we  shall  have  x^-x'^  =  x'^,  or  3:3=2^2  ;  and  con- 
sequently a:=32,  and  l+2x  —  x'^-\-x^=z\  +4  —  4  +  8=r9,  a 
square  as  required. 

Ex.  2.  Find  such  a  value  of  x  as  will  make  the  expression 
3x^  —  5x'^  +  6x+4:  a  square. 

Put  3a;3-5a:2+6a:+4  =  (fx-f-2)2  =  |a;2  +  6x+4, 
then,  3x3-5a;2=:fx2,  or  3x-5  =  |;    .-.  a:=?f ,  which  being 

substituted  in  the  proposed  expression,  makes  it  equal  to  [-^j  ^ 

PROBLEM  III. 

544.  To  render  surd  quantities  of  the  form  y/{a'\-bx'\-cx'^ 
-\-dx^-{-ex'^),  rational,  or  to  find  such  values  of  x  as  will  make 
a-\-bx-\'Cx'^-{'dx^-{-ex^  a  square. 

RULE. 

Case  1.  When  o  is  a  square  number,  put  it  =/2,  and  make 
p  +  bx  +  cx''-\-dx^-i-ex^=.{f+yx+^^^-^^x^)  =P+bx-{- 

^^  + 87^""  "^^  +  ""64/^         ' 

then  since  the  first  three  terms  on  each  side  of  the  equation 

destroy  each  other,  we  shall  have  — ^-pr^ —  a;*i ^g^^^ 

x'^  =  ex^-{-dx'^ ;  and  therefore,  by  division  and  reduction, 
64dp-8bpi4cp-b^ 

^~     {'icf^—bY  —  ^'^cp      ' 
which  form  fails  when  any  two  of  the*  coefficients  b,  c,  d,  are 
each  =0. 

Example  1.  It  is  required  to  find  such  a  value  of  x,  that 
l—2x-\-2x-  —  Ax^-\-bx^  shall  be  a  square. 

Here,  the  first  term  being  a  square  number,  let  1— 2a;4-3a;2 

4a:3+5a;*=(l-a;— 1:2)2=.  1—2x4-3x2 -2a:3  +  x*. 

Then,  since  the  first  three  terms  on  each  side  of  the  equa- 
tion destroy  each  other,  we  shall  have  5a:*— 4a;=^=x*— 2x3; 


364  DIOPHANTINE  ANALYSIS. 

.'.x=^,  and  consequently  l—2x-\-3x^—4x^-^5x^=:zl  —  l-^^ 
— 5+tV— tV  »  which  is  a  square  number,  as  was  required. 
Ex.  2.  Find  such  a  value  of  x  that  we  may  have  22x*— 40a;3 

o 

— 40a;^+64a:4-16  a  square.  Ans.  ~. 

545.  Case  2.  When  c  is  a  square  number,  put  it  =_g-2,  and 

mdkeg^x*-\-dx^-\-cx'^-\-bx+a=(gx'^-{ x+   %T    )  =g^^* 

+c?a;3  4-ca;2+^^-!p^!l^+il!^^=^!l!  ;  then,  we  shall  have 


(4c^2_^2y2_64a^6 


64^6—+        Qj. 


646o-6_8%2(4c^2_^2)- 

which  form  also  fails  in  the  same  case  as  the  former. 

Example  1 .  It  is  required  to  find  such  a  value  of  Xj  that 
— 2-\-2x — x'^—2x^-\-4:X^  shall  be  a  square  number. 

Let4a:*-2a:3— a;24-3a:— 2=(2aj2_ia,__5_)2^4^4_2a;3-a;2 
+  T6^+2¥¥  »  ^^^"^  '^^^   shoW.  have   "6x—2=^x-\-^-fQ\   .'.x 

537 

' —  688* 

Ex.  2.  It  is  required  to  find  such  a  value  of  x,  that  4a;*+ 
4x^-{'4x'^-\-2x — 6  shall  be  a  square. 

Vut  4x^-\-4x^'\-4x^-h2x-e  =  {2x^  +  x-h^f  = 

4x'^-{-4x^-\-4x^~\-%x^-fQ,  and  we  "have 
2x-6^?,x+j%;  .'.x=.l3l. 
546.  Case  3.  When  the  first  and  last  terms  are  both  squares, 
put  a=f^  and  e^ig"^,  and  make  f^-{-bx-{-cx'^-\-dx^-\-g^x*= 

(/  +  ^a:+^a:2)2^y-2+j^+(2/o-+^^)2^24.^^34.^,2^4. 

then,  since  the  second  terms,  as  well  as  the  first  and  last,  on 
each  side  of  the  equation,  destroy  each  other,  we  shall  have 

f(bg-fd)    '■ 
And  because  g  is  found  in  the  original  formula  only  in  its 
second  power,  it  may  be  taken  either  positively  or  negatively ; 
and  consequently  we  shall  also  have 

,_i*!-/W^_±£). 

■o  that  this  mode  of  solution  furnishes  two  diflferent  answers. 


DIOPHANTINE  ANALYSIS.  365 

Example  1.  It  is  required  to  find  such  a  value  of  a?,  as 
shall  make  l-{'3x-\-7x'^—2x^-\'4x*  a  square. 
Let  l  +  3a;+7x2—2x3  +  4ic4  =  (l+fx-f  2x2)2== 

25 
l  +  3i:+— a:2-f6a;3+4a;*; 
4 

25  3 

.-.  6a:3+— a:2=:7a;2-.2a;3,  and  x=—. 
4  o<^ 

Ex.  2.  It  is  required  to  find  such  a  value  of  of,  as  shall  make 

16— 24a? +4 Jc^  — 6x3  4- a;*  ^  square. 

Letx*— 6a:3  +  4a;2-24a;-}-16  =  (a:2— 3a:— 4)2= 
x*—6x^-\-x^+24x-{-\6, 
and  there  results 

4x2— 24x=a:2^24x,  or  4x— 24=a?+24  ; 
.•.x=l6. 

PROBLEM  IV. 

547.  To  render  surd  quantities  of  the  form  ^  {a-\-bx-{-cx'^ 
-\-dx^)  rational,  or  to  find  such  values  of  x  as  will  make  a-\- 
bxi-cx'^-\-dx^  a  cube. 

Case  1.  When  a  is  a  cube   number,  put  it  =e^,  and  take 

h  ^2  ^3 

e3j^bx+cx^-\-dx^={e+—-xy=e^-{-bx-\-:^x^  +  ^;^x^;ihen 

b^  ^2 

we  shall  have  6?a?3-f-cx2=— -x^-f-— -x2 ;  or,  by  dividing  by 

x2,  and  reducing  the  terms, 

9e3(3ce3— 52) 
21[de^x-^21ce^=.¥x-\-9h'^e^ :  whence  x=.-^ — -'. 

Example  1.  It  is  required  to  find  such  a  value  of  x,  as  will 
make  the  formula  l+x-4-x2  a  cube. 

Let  l-fx-|-x2  =  (l+^x)3  =  H-x-|-lx2+^Lx3,  or 
x2z=:ix24-2V^^  »  •'•  3:  =  18,  and  consequently, 
l+a?+a:2=l  +  18+324  =  343  =  73,  a  cube  number,  as   was 
required. 

Ex.  2.  It  is  required  to  find  such  a  value  of  x  that  will 
make  the  formula  2x-^  +  3x2  — 4x+8  a  cube. 

Let2x3  +  3x2-4x+8  =  (— Jx+-2)3=  — 2T*3+fx2— 4x+8, 
and  we  have  2x3-f  3x2=:— Jyx34-|x2, 

or  2x4-3= -Jy^+f; 
.-.  x=-fj. 
Ex.  3.  It  is  required  to  find  such  a  value  of  x,  as  to  make 
the  formula  3x34-2x4-1  a  cube.  Ans.  ^. 

548.  Case  2,  When  c?  is  a  cube   number,  put  it  =/3,  and 

32* 


366  DIOPHANTINE  ANALYSIS. 

take   a^hx+cx^-\-px^  =  (_i_+/,,)3^_^+^^  4.  ,^2  ^ 

j^ac^  ;  then    we   shall  have    a  +  ^a;=  — -—-\-—x  ;    .•.  a;=: 

27/^      3/3 

27ff/6-c3 
9/3(c2-3i/^)' 

Example  1.  It  is  required  to  find  such  a  value  of  a:  as  will 
make  ISS  +  Sx^-fa:^  a  cube. 

Let  1334-3x2+a:3  =  (14-a:)3=l-f  3a:  +  3a;2-f.a;3;  and  since 
the  two  last  terms  of  this  equation  destroy  each  other,  there 
will  remain  1  +  3j:=:133,  or  3a;=133  — 1  =  132  ;  whence  a;  = 
i|2r=44,  and  consequently  1334-3a;2+a;3=92025  =  (45)3,  a 
cube  number,  as  was  required. 

Ex.  2.  It  is  required  to  find  such  a  value  of  a;  as  will  make 
the  formula  Sa:^  — 4a;2-|-2a  — 12  a  cube. 

Let  Sx^  -  4a:2+2a;  -  12  =  (2a:  -  ^f^^x'^-Ax'^+lx-^^, 
and  we  have  2x—\2  —  '^x-\--^-j\ 

.     -, 325 

.  .  X—   3g^  . 

Ex.  3.  It  is  required  to  find  such  a  value  of  a?  as  will  make 
the  formula  a:^  — 3a;2-f  a;  a  cube.  Ans.  x=^. 

549.  Case  3.  When  a  and  d  are  both  cube  numbers,  let 
them  be  put  z=ze^  and/^^  and  make  e^-{-bx-\-cx^-\-f^x^=:[e-\- 
fxYz=ie'^-\-3fe'^x-{-^ef'^x^'-\-f^x'^  \  then,  we  shall  have  hx^cx"^ 

=  ^fe^x-r^ef^x'^  ;  /.  xz=i- — ^^2»  which  formula  may  be  also 
c — 6ej 

resolved  by  either  of  the  two  first  cases. 

Example  1.  It  is  required  to  find  such  a  value  of  a?,  that 
8  +  28a;  +  89x2  — 125a;3  shall  be  a  cube. 

Let  8  +  28a:-f89x2-125a:3z:z(2-5a:)3z=8-60a:+150a;2— 
125a:3  .  and,  since  the  first  and  last  terms  of  this  equation 
destroy  each  other,  there  will  remain  28a:4-89a:2= — 60a:-f- 
150a;2;  .-.  150a;— 89a:=28  +  60,  or,  61a:=88,  and  a:=-||,  the 
value  required.  And  as  this  formula  can,  also,  be  resolved 
by  the  first  or  second  case,  other  values  of  x  may  be  obtained, 
that  will  equally  answer  the  conditions  of  the  question. 

Ex.  2.  It  is  required  to  find  such  a  value  of  x,  that  the  for- 
mula S-\-Ax-\-^x'^-\-x'^  shall  be  a  cube. 

Let  8  +  4a;+9a:2+a:3=(2H-ar)3=^8+12a:+6a;2+a:3,  and  we 
shall  then  have  9x2  +  4a:=6a:2+12a; ;  .-.  a:  =  2f. 

PROBLEM   V. 

On  the  Resolution  of  Double  and  Triple  Equalities. 

550.  When  a  single  formula,  containing  one  or  more  un- 
known quantities,  is   to  be  transformed  to  a  perfect  power, 


DIOPHANTINE  ANALYSIS.  367 

such  as  a  square  or  a  cube,  this  is  called  in  the  Diophantine 
Analysis,  a  simple  equality  ;  and  when  two  formulae,  con- 
taining the  same  unknown  quantity,  or  quantities,  are  each 
to  be  transformed  to  some  perfect  power,  it  is  then  called  a 
double  equality,  and  so  on  ;  the  methods  of  resolving  which, 
in  such  cases  as  admit  of  any  rule,  are  as  follows. 

Prob.  1.  When  the  unknown  quantity  does  not  exceed 
the  first  degree,  as  in  the  double  equality 

a-\-hx:=  a  square,  and  c^dxz=z  a  square. 

Let  the  first  of  these  formula  a-f-^a?— ^^,  and  the  second 
c-\-dx=.n^  ;  then,  by  eliminating  x  from  each  of  these  equa- 
tions, we  shall  have  hu'^-\-ad  —  bc=zdt^,  or  bdu'^-{-{ad—bc)d  = 
d'^t^  ;  and  since  the  quantity  on  the  right  hand  side  of  this 
last  equation  is  now  a  square,  it  is  only  necessary  to  find  such 
a  rational  value  of  m,  as  will  make  bdu^-\-{ad  —  bc)d  a  square, 
which  being  done  according  to  one  of  the  methods  already 

explained,  we  shall  have  xz=  — - — . 

Example.  It  is  required  to  find  a  number  a;,  such,  that  x-\- 
128  and  a:+192  shall  be  both  squares. 

Here,  let  a:+ 128=: w^  and  a:-f  192  =  ^2  .     • 

then,  by  eliminating  x,  we  shall  have  w^  — 128  =  ^2 — ^92  ;  or 
w2-j-64z=^2.  and,  as  the  quantity  on  the  right-hand  side 
of  the  equation  is  now  a  square,  it  only  remains  to  make 
tt2-f-64  a  square  ;  for  which  purpose,  put  u'^-{-64=:{u-\-nY  = 

64  — n^ 
u'^-^2nu-{-n^,  2nu+n'^  =  64  ;  whence,  u=—— —  ;  or,  taking 

64  —  4 
n,  which  is  arbitrary,  =2,  we  shall  have  u= — - — =15  ;  and 

consequently,  x  =  7i2— 128=225  — 128  =  97,  the  number  re- 
quired. 

551.  Prob.  2.  When  the  unknown  quantity  does  not  ex- 
ceed the  second  degree,  and  is  found  in  all  the  terms  of  the 
two  formulae,  as  in  the  double  equality,  ax'^-\-bx=  a  square, 
and  cx^+dx=  a  square. 

Let  x=-  ;  then,  by  multiplying  each  of  the  two  resulting 

equations  by  y"^,  we  shall  have 

a-\-byz=i  a  square,  and  c-{-dy=  a  square  ; 
from  which  the  value  of  y,  and  consequently  that  of  x,  may 
be  determined,  as  in  Problem  L 

But  if  it  were  required  to  transform  the  two  general  expres- 


368  DIOPHANTINE  ANALYSIS. 

sions  <z+Jj;-}-ca:'^  and  d-{-ex-\-fx^  into  squares;  the  solution 
could  only  be  obtained  in  a  few  particular  cases,  as  the  result- 
ing equality  would  rise  to  the  fourth  power. 

ExAiMPLE.    It  is  required  to  find  a  number  x,  such,  that 
ar^-f-x  and  x^—x  shall  be  both  squares. 

Here,  let  xz=:-;  then  the  two  formulae  in  the  question  will 

y 

become  — --j —  and  -— ,  or  — ::(l  +  v)  and  -^(1— y),  which 

are  to  be  squares.  But  since  a  square  number,  when  multi- 
plied or  divided  by  a  square  number,  is  still  a  square,  it  is  the 
same  thing  as  to  transform  1  +y  and  1  — y  to  squares  ;  for 
which  purpose,  let  l-fy=j3^,  or  i/=zp^—l  ;  \—i/z=z2—p'^j 
which  is  also  to  be  a  square.  But  as  neither  the  first  nor  last 
terms  of  this  new  formula  are  squares,  we  must,  in  order  to 
succeed,  find  some  simple  number,  that  will  answer  the  con- 
dition required  ;  which,  it  is  evident,  from  inspection,  will  be 
the  case  when/)=rl. 

Let,  therefore,  p  =  1—5',  and  we  shall  have  i—r/z=z2-~p^  = 
]-}-2q—q^;  or,  putting  l+2q—q^  =  {l—rqy-=l—2rq+rYi 

2r-\-2 
whence  2'—q:=-^r-\-r^q,  or  q,'=—^ — r;  and   consequently, 

^1^ 1      ^(l+r)\ 

^     y     t—^9.     4r— 4H' 
where,  in  order  to  make  x  positive,  r  may  be  taken  equal  to 
any  proper  fraction  whatever. 

Let,  therefore,  for  the  sake  of  greater  simplicity,  r=:-,  and 

we  shall  have  a;  ~        ,  ^ — ^;    in    which    case,  any   whole 

numbers  may  be  now  substituted  for  u  and  ^,  provided  «  be 

greater  than  t. 

^5 
If,  for  instance,  w=2  and  ^=1,  we  shall  have  a^=.v7  I  and 

169 
if  w  =  S  and  f =2,  a;=— ^  ;  and  so  on,  for  any  other  numbers. 

552.  pROB.  3.  In  the  case  of  a  triple  equality,  where 
ihree  expressions  of  the  former,  ax-\-hy^  cx-\-dj/,  and  ex-^fy 
are  to  be  transformed  to  squares. 

Let  the  first  of  them  ax-^by^t"^,  the  second  cx-^dy=u^f 
d  the  third  ex-\-fy=:s^ ;  then,  if  x  be  eliminated  from  each 


DIOPHANTINE  ANALYSIS.  369 

of  these  equations,  and  afterwards  y  in  the  two  resulting 
equations,  we  shall  have  {af—he)u*^i-{cf—de)i'^-==.{ad--c}))s'^  • 

and  by  putting  -==z,  or  u=:tz,  there  will  arise 

af — he      ^     cf —  de      s^ 

ad — cb  '  ad —  cb       t^  * 

and  since  the  quantity  on  the  right-hand  side  of  this  equation, 
is  a  square,  it  only  remains  to  find  such  a  rational  value  of  z 
as  will  make 


of— be    _     cf—de 

~ -z^ — ~- r  a  square 

ad — CO  ad — cb 


which  being  done  by  one  of  the  methods  before  explained, 
we  shall  readily  obtain,  by  means  of  the  first  two  equations, 
d—bz"^  o         1         az'^  —  c  „ 

where  t  may  be  any  number  whatever. 

Example.  It  is  required  to  find  three  numbers  in  arith- 
metical progression,  such,  that  the  sum  of  every  two  of  them 
may  be  a  square. 

Let  X,  x-\-y,  x-{-2i/=:  the  three  numbers  ;  and  put  2x-\-y=i 
^2,  2x-{-2i/=u-,  and  2x-\-3i/=^s'^  ;  then,  by  eliminating  a;  and 
y  from  these  equations,  we  shall  have  u^ — 1^  =  3"^— u"^,  or  2u^ 
—  t'^=zs'^ ;  and  if  we  now  put  u  =  tZj  then  will  arise  2t^z^' — t"^ 

=  ^2,  or  2z'^—\=—  ;  where,  —  being  a  square,  it  only  re- 
mains to  make  2^  —  1  a  square;  what  it  evidently  is  when 
z:=:z\.  But  as  valuc  would  be  found  not  to  answer  the  con- 
ditions of  the  question,  let  ^rrrl— jd  ;  2z'^—l=2{l — p)'^—l=z 
1— 4;?  +  2p2;  and  by  putting  this  last  expression  —{l—rp)^, 
we   have    l—4p+2p'^  —  l  —  2rp-\-r^p^,  or  — 4  +  2/)  =  — 2rH- 

2r— 4         ,         ,      2r-4     r^— 2r-f2 
ry  \  whence  p=-2- 


,  .  m. 

making  r=—,  z—  002 

And  since,  by  the  first  two  equations,  yz::zu^—t'^=:t'^z'^  —  t^ 
=  (^2-1)^2^  and  a;=i(z2_y2)_^(2-;22)^2  .  it  is  evident,  that 
z  must  be  some  number  greater  than  1  and  less  than  'v/2. 

,    „x.  81—90  +  50 

If,  therefore,  7n=9  and  7i=5,  we  shall  have  z=. —  ~ — 

41  241      ^2  720      .  *T      ,     ovvQi 

=  3j-,  a:=— -X-,  andy=— -Xi2;  or,  takmg/=2  x31,  a;= 

482,  andy=2880.     Hence, 


370  DIOPHANTINE  ANALYSIS. 

a;  — 482,  a:+y  =  3362,  and  a;4-2y=6242, 
the  numbers  required.    #1 

EXAMPLES  FOR  PRACTICE. 

Ex.  1.  It  is  required  to  find  a  number  such,  that  x-{-l  and 
X — 1  shall  be  both  squares.  Ans.  a:=|. 

Ex.  2.  It  is  required  to  find  a  number  x,  such,  that  a;-f-4 
and  x-{-7  shall  be  both  squares.  Ans.  xz=^^. 

Ex.  3.  It  is  required  to  find  two  numbers  such,  that  if  their 
product  be  added  to  the  sum  of  their  squares,  the  result  shall 
be  a  square.  Ans.  3  and  5. 

Ex.  %.  Find  two  numbers  such,  that  if  the  square  of  each 
be  added  to  their  product,  the  sums  shall  be  both  squares. 

Ans.  9  and  16. 

Ex.  5.  It  is  required  to  find  two  whole  numbers  such,  that 
the  sum  or  difference  of  their  squares  when  diminished  by 
unity,  shall  be  a  square.  Ans.  8  and  9. 

Ex.  6.  To  find  two  whole  numbers  such,  that  if  unity  be 
added  to  each  of  them,  and  also  to  their  halves,  the  sums  in 
both  cases  shall  be  squares.  Ans.  48  and  1680. 

Ex-  7.  It  is  required  to  find  three  square  numbers,  that 
shall  be  in  arithmetical  progression.  Ans.  1,  25,  and  49. 

Ex.  8.  It  is  required  to  find  three  square  numbers  that  shall 
be  in  harmonical  proportion.  Ans.  1225,  49  and  25. 

Ex.  9-  To  find  three  whole  numbers  such,  that  if  to  the 
squaro'-of  each  the  product  of  the  other  two  be  added,  the 
sums  shall  be  squares.  Ans.  9,  73  and  328. 

Ex.  10.  It  is  required  to  resolve  4225,  which  is  the  square 
of  65,  into  two  other  integral  squares.       Ans.  2704  and  1521. 

Ex.  11.  It  is  required  to  resolve  9^+22,  or  85  into  two 
other  integral  squares.  •  Ans.  1*^-^6'^. 

Ex.  12.  It  is  required  to  find  three  square  numbers,  such, 
that  their  sum  shall  be  a  square.  Ans.  9,  16  and  \f^. 

Ex.  13.  To  find  two  numbers  such,  that  their  sum  shall' be 
equal  to  their  cubes.  Ans.  ^  and  J 


371 


APPENDIX 


Algebraic  Method  of  demonstrating  the  Propositions  in  the  fifth 
book  of  Euclid'' s  Elements^  according  to  the  text  and  arrange- 
ment  in  Simson^s  edition. 

Simson's  Euclid  is  undoubtedly  a  work  of  great  merit,  and 
is  in  very  general  use  among  mathematicians  ;  but  notwith- 
standing all  the  efforts  of  that  able  commentator,  the  fifth  book 
still  presents  great  difficulties  to  learners,  and  is  in  general  less 
understood  than  any  other  part  of  the  Elements  of  Geometry. 
The  present  essay  is  intended  to  remove  these  difficulties,  and 
consequently  to  enable  learners  to  understand  in  a  sufficient  de- 
gree the  doctrine  of  proportion,  previously  to  their  entering 
on  the  sixth  book  of  Euclid,  in  which  that  doctrine  is  indis- 
pensable. 

I  have  omitted  the  demonstrations  of  several  propositions, 
which  are  used  by  Euclid  merely  as  lemmata,  but  are  of  no 
consequence  in  the  present  method  of  demonstration. 

Instead  of  Euclid's  definition  of  proportion,  as  given  in  his 
fifth  definition  of  the  fifth  book,  I  make  use  of  the  common  al- 
gebraic definition  ;  but  I  have  shown  the  perfect  equivalence 
of  these  two  definitions.  This  perfect  reciprocity  between  the 
two  definitions  is  a  matter  of  great  importance  in  the  doctrine 
of  proportion,  and  has  not  (as  far  as  I  can  learn)-been  discuss- 
ed by  any  preceding  mathematician. 

With  respect  to  compound  ratio,  I  have  also  given  another 
definition  of  it  instead  of  that  given  by  Dr.  Simson  ;  as  his 
definition  is  found  exceedingly  obscure  by  beginners,  and  is 
in  my  judgment  one  of  the  most  objectionable  things  in  his 
edition  of  Euclid's  Elements. 

The  literal  operations  made  use  of  in  the  present  paper  are 
extremely  simple,  and  require  very  little  previous  knowledge 
of  algebra  to  render  them  intelligible. 

The  algebraic  signs  commonly  used  to  indicate  greater^ 
equal,  less,  are  7,  =,  Z  -  '^hus  the  three  expressions  a/' 6 
c=d,  eZf  signify  that  a  is  greater  than  b,  that  c  is  equal  to  d^ 
and  that  e  is  less  than  f.  The  expression  c=J  is  called  an 
equation  or  equality;  the  others  a  76,  e^/,  are  called  in- 
equalities. 


372         .  APPENDIX. 

Also  when  four  quantities  are  proportionals,  we  shall  express 
this  relation  in  the  usual  mode  by  points  ;  thus, 

A  :  B  : :  C  :  D 
is  to  be  read,  A  is  to  B  as  C  is  to  D  ;  or,  A  has  the  same  ratio 
to  B  that  C  has  to  D. 

THE  ELEMENTS  OF  EUCLID,  BOOK  V. 

Definitions. 
I. 
A  less  magnitude  is  said  to  be  a,  part  of  a  greater,  when  the 
less  measures  the  greater,  that  is,  when  the  less  is  contained  a 
certain  number  of  times  exactly  in  the  greater. 

II. 
A  greater  magnitude  is  said  to  be  a  multiple  of  a  less,  when 
the  greater  is  measured  by  the  less,  that  is,  when  the  greater, 
contains  the  less  a  certain  number  of  times  exactly. 

III. 
Ratio  is  a  mutual  relation  of  two  magnitudes  of  the  same 
kind  to  one  another  in  respect  to  quantity. 

IV. 
Magnitudes  are  said  to  have  a  ratio  to  one  another,  when 
the  less  can  be  multiplied  so  as  to  exceed  the  other. 

V. 
The  ratio  of  the  magnitude  A  to  the  magnitude  B  is  the 
number  showing  how  often   A  contains  B  ;  or,  which  is  the 
same  thing,  it  is  the  quotient  when  A  is  numerically  divided  by 
B,  whether  this  quotient  be  integral,  fractional,  or  surd. 
Explication. 
This  fifth  definition,  with  its  corollaries,  is  used  in  the  pre- 
sent essay  instead  of  Euclid's  5th  and  7th  definitions  :  the 
following  examples. Avill  sufficiently  illustrate  the  definition. 
Let  A =20,  and  Br=5,  then  the  ratio  of  A  to  B,  or  of  20  to  5, 

A       20 
is  -^  ^^~r^  ^^  ^'  ^^  '^^^  ^^®  ratio,  of  20  to  5  is  4.     Again,  let 

xJ  O 

A       5      1 
A=5,  and  B::=20,  then  .^  =  --  =  -,  and  therefore  the  ratio  of 
Jj       20     4 

1  A 

5  to  20  is-      Lastly,  let  A  =  12^2,  and  B=4,  then  ■^= 
4  D 

Pv^_.3   /2^  and  therefore  the  ratio  of  12^2  to  4  is  3^2. 
4 
Corollary  I.  If  four  magnitudes,  A,  B,  C,  D,  be  so  related 

A      C 
that  -=r~=r=-,  it  is  evident  the  ratio  of  A  to  B  is  the  same  with 
D        D 

the  ratio  of  C  to  D. 


APPENDIX.  373 

Cor.  TI.  Any  four  magnitudes  whatever,  so  related  that  the 
ratio  of  the  first  to  the  second  is  the  same  with  the  ratio  of  the 
third  t«rthe  fourth,  may  be  expressed  by 

rA,  A,  rB,  B  ; 
the  first  of  the  four  being  rA,  the  second  A,  the  third  rB,  and 
the  fourth  B  ;  the  magnitudes  A  and  B  being  any  whatever, 
and  the  letter  r  denoting  each  of  the  two  equal  ratios  or  quo- 
tients when  the  first  rA  is  divided  by  the  second  A,  and  the 
third  rB  divided  by  the  fourth  B. 

Cor.  III.  When  four  magnitudes  A,  B,  C,  D,  are  so  relat- 

A  C 

ed  that  -^  is  greater  than  — ,  it  is  evident  that  the  ratio  of  A 

to  B  is  greater  than  the  ratio  of  C  to  D  ;  or  t*hat  the  ratio  of  C 
to  D  is  less  than  the  ratio  of  A  to  B.  •     . 

The  Fifth  Definition  a^^ding  to  Euclid. 

The  first  of  four  magnitudes  is  said  to  have  the  same  ratio 
to  the  second  which  the  third  has  to  the  fourth,  when  any 
equimultiples  whatsoever  of  the  first  and  third  being  taken, 
and  any  equimultiples  whatsoever  of  the  second  and  fourth, 
if  the  multiple  of  the  first  be  less  than  that  of  the  second,  the 
multiple  of  the  third  is  also  less  than  that  of  the  fourth  ;  or, 
if  the  multiple  of  the  first  be  equal  to  that  of  the  second,  the 
multiple  of  the  third  is  also  equal  to  that  of  the  fourth  ;  or  if 
the  multiple  of  the  first  be  greater  than  that  of  the  second, 
the  multiple  of  the  third  is  also  greater  than  that  of  the 
fourth. 

Scholium.  We  shall  demonstrate  towards  the  close  of  this 
essay,  that  this  definition  of  Euclid's  and  our  5th  definition, 
acccording  to  the  common  algebraic  method,  are  not  only  con- 
sistent with  each  other,  but  also  perfectly  equivalent,  each 
comprehending,  whatsoever  is  comprehended  by  the  other. 

VI. 

When  four  magnitudes  are  proportionals,  it  is  usually  ex- 
pressed by  saying,  the  first  is  to  the  second  as  the  third  to  the 
fourth. 

The  Seventh  Definition  according  to  Euclid. 

When  of  the  equimultiples  of  four  magnitudes,  (taken  as 
in  the  fifth  definition)  the  multiple  of  the  first  is  greater  than 
that  of  the  second,  but  the  multiple  of  the  third  is  not  greater 
than  that  of  the  fourth ;  then  the  first  is  said  to  have  to  tne 
second  a  greater  ratio  than  the  third  has  to  the  fourth  ;  and, 
on  the  contrary,  the  third  is  said  to  have  to  the  fourth  a  less 
ratio  than  the  first  has  to  the  second. 

33 


374  APPENDIX. 

VIII. 
Analogy  or  proportion  is  the  equality  of  ratios. 

IX.  ^  • 

Omitted. 
X. 
When  three  magnitudes  are  proportionals,  the  first  is  said  to 
have  to  the  third  the  duplicate  ratio  of  that  wliich  it  has  to  the 
second. 

XL 
When  four  magnitudes  are  continued  proportional,  the  first 
is  said  to  have  to  the  fourth  the  triplicate  ratio  of  that  which 
it  has  to  the  secpnd,  and  so  on,  quadruplicate,  &c.  increasing 
the  denomination  still  by  unity  in  any  number  of  propor- 
tionals. 

Definition  A,  viz.  of  cj^ound  ratio,  omitted. 

WXII. 
In  proportionals,  the  antecedent  terms  are  called  homologous 
to  one  another,  as  also  the  consequents  to  one  another. 

XIII. 
Permutando,  or  Alternando,  by  permutation,  or  by  alter- 
nation, or  alternately,  are  terms  used,  when  of  four  propor- 
tionals it  is  inferred  that  the  first  is  to  the  third  as  the  second 
*  to  the  fourth. 

XIV. 
Invertendo  by  inversion,  or  inversely,  when  of  four  propor- 
tionals, it  is  inferred  that  the  second  is  to  the  first  as  the  fourth 
to  the  third. 

XV. 
Componendo,  by  composition,  when  it  is  inferred  that  the 
sum  of  the  first  and  second  is  to  the  second  as  the  sum  of  the 
third  and  fourth  is  to  the  fourth. 

XVI. 
Dividendo,  by  division,  when  it  is  inferred  that  the  excess 
of  the  first  above  the  second  is  to  the  second  as  the  excess  of 
the  third  above  the  fourth  is  to  the  fourth. 

XVII. 
Convertendo,  by  conversion,  or  conversely,  when  it  is  in- 
ferred that  the  first  is  to  its  excess  above  the  second,  as  the 
third  to  its  excess  above  the  fourth. 
XVIII. 
Ex  aequali  (sc.  distantia),  or  ex  aequo,  from  equality  of  dis- 
tance, when  there  is  any  number  of  magnitudes  more  than 
two,  and  as  many  others,  so  that  they  are  proportionals  when 
taken  two  and  two  of  each  rank,  and  it  is  inferred  that  the  first 
18  to  the  last  of  the  first  rank  of  magnitudes  as  the  first  is  to 


APPENDIX.  375 

the  last  of  the  others  :  of  this  there  are  the  two  following 
kinds,  which  arise  from  the  different  order  in  which  the  mag- 
nitudes are  taken  two  and  two. 

XIX. 

Ex  aequali,  from  equality;  this  term  is  used  simply  by  it- 
self, when  the  first  magnitude  is  to  the  second  of  the  first 
rank,  as  the  first  to  the  second  of  the  other  rank,  and  the  se- 
cond to  the  third  of  the  first  rank  as  the  second  to  the  third 
of  the  other;  and  so  on  in  order;  and  it  is  inferred  that  the 
first  is  to  the  last  of  the  first  rank  as  the  first  is  to  the  last  of 
the  other  rank, 

XX. 

Ex  asquali,  in  proportione  perturbata,  seu  inordinata,  from 
equality  in  perturbale  proportion  ;  this  term  is  used  when 
the  first  is  to  the  second  of  the  first  rank  as  the  last  but  one 
to  the  last  of  the  other  rank,  and  the  second  is  to  the  third  of 
the  first  rank  as  the  last  but  two  to  the  last  but  one  of  the 
other  rank,  and  so  on  in  a  cross  order ;  and  it  is  inferred  that 
the  first  is  to  the  last  of  the  first  rank  as  the  first  is  to  the  last 
of  the  other  rank. 

XXI. 

If  A,  B,  C,  D,  be  any  number  of  magnitudes  of  the  same 
kind,  and  P  any  other  magnitude  ;  and  if  we  make  A  :  B  :  : 
P  :  Q  ;  and  B  :  C  :  :  Q  :  R ;  and  C  :  D  :  :  It  :  S  ;  the  ratio 
of  P  to  S  is  said  to  be  compounded  of  the  ratios  of  A  to  B,  B 
to  C,  C  to  D. 

AXIOMS. 

I.  Equimultiples  of  the  same,  or  of  equal  magnitudes,  are 
equal. 

II.  These  magnitudes  of  which  the  same,  or  equal  magni- 
tudes, are  equimultiples,  are  equal  to  one  another. 

III.  A  multiple  of  a  greater  magnitude  is  greater  than  the 
same  multiple  of  a  less, 

IV.  That  magnitude  of  which  a  multiple  is  greater  than 
the  same  multiple  of  another,  is  greater  than  that  other  mag- 
nitude. 

PROPOSITIONS. 

Propositions  I.  II.  III.  V.  and  VI.  are  omitted,  as  they  do 
not  treat  of  proportion,  and  are  not  wanted  in  the  method  of 
demonstration  adopted  in  this  essay. 

PROP.  IV.  THEOR. 

If  the  first  of  four  magnitudes  has  the  same  ratio  to  the  se- 
cond which  the  third  has  to  the  fourth ;  then  any  equimulti- 
ples whatever  of  the  first  and  third  shall  have  the  same  ratio 
to  any  equimultiples  of  the  second  and  fourth  ;  that  is,  the 


376  APPENDIX. 

equimultiple  of  the  first  shall  be  to  that  of  the  second  as  the 
equimultiple  of  the  third  is  to  that  of  the  fourth. 

DEMONSTRATION. 

By  Cor.  2.  Def.  5.  let  any  four  proportionals  be  repre- 
sented by 

rA,  A,  rB,  B  ; 
and  m  and  n  being  any  two  integers  greater  than  unity,  the 
equimultiples  of  rA  and  rB  will  be 

m/*A,  mr^  ; 
and  in  like  manner  the  equimultiples  of  A,  B,  will  be  wA,  nB. 
We  are  to  prove  that   the   four  following  quantities,  mrA, 
wA,  TwrB,  7iB,  are  proportionals. 

171T  P^       ftlT 

Bv  Def.  5.  the  ratio  of  mrA  to  nk  is 


nk 

'  n 

mrB 

m? 

nB~ 

n 

and  the  ratio  of  mrB  to  nB  is  — rr  = 

TflT 

now  these  two  ratios  being  each  =  — , 

n 

are  manifestly  equal  to  each  other,  and  therefore  By  Cor.  1. 

Def.  5. 

mrA  :  nA  :  :  mrB  :  ?iB.  Q.  E.  D. 

CoR.  Likewise  if  the  first  be  to  the  second  as  the  third  to 

the  fourth,  then  also  any  equimultiples  of  the  first  and  third 

shall  have  the  same  ratio  to  the  second  and  fourth ;  and,  in 

like  manner,  the  first  and  third  shall  have  the  same  ratio  to  any 

equimultiples  of  the  second  and  fourth. 

DEMONSTRATION. 

We  have  first  to  prove  that  the  four  following, 

mrAj  A,  mrB,  B  are  proportionals. 

mi  •       /.       .         .   .    mrA 

The  ratio  of  mrA  to  A  is  -—-=mr, 
B 

and  the  ratio  of  mrB  to  B  is  -^—=mr ; 

Therefore  mrA  :  A  : :  mrB  :  B. 
In  like  manner  we  prove  that  rA  :  nA  : :  rB  :  nB. 

PROP.  A.   THEOR. 

If  the  first  of  four  magnitudes  has  the  same  ratio  to  the 
second  which  the  third  has  to  the  fourth  ;  then  if  the  first  be 
greater  than  the  second,  the  third  is  also  greater  than  the 
fourth  ;  if  equal,  equal ;  and  if  less,  less. 

DEMONSTRATION. 

By  Cor.  1.  Def.  5.  any  four  proportionals  maybe  expressed 
by 

rA,  A,  rB,  B. 


APPENDIX.  377 

If  we  have  rA/' A,  J  ifrA=A,  J  ifrA^^A,  J 
then  by  division  r/ij  /  then  r=l,  >  then  r/ 1,  > 
and  by  multip.     rB/B,  )  and  rB=:B,  )  and  rB/B.  > 

Q.  E.  D. 

PROP.  B.  THEOR. 

If  four  magnitudes  are  proportionals,  they  are  proportionals 
also  when  taken  inversely. 

DEMONSTRATION. 

Let  rA,  A,  rB,  B  be  any  four  proportionals,  we  are  to  prove 

that  A,.rA,  B,  rB  will  also  be  proportionals. 

A      1 

The  ratio  of  A  to  rA  is  —r-=- 

rA     r 

t)       1 

and  the  ratio  of  B  to  rB  is  -7^=-  ; 
rtJ     r 

and  therefore 

A  :  rA  :  :  B  :  rB.  Q.  E.  D. 

PROP.  C.  THEOR. 

If  the  first  be  the  same  multiple  of  the  second,  or  the  same 
part  of  it  that  the  third  is  of  the  fourth  ;  the  first  is  to  the 
second  as  the  third  is  to  the  fourth. 

DEMONSTRATION. 

1 .  Supposing  m  to  be  any  integer  greater  than  unity,  let  mA 
the  first  be  the  same  multiple  of  the  second  A,  that  mB  the 
third  is  of  the  fourth  B  ;  we  are  to  prove  that  mA,  A,  wiB,  B 
are  proportioi^ls. 

The  ratio  of  mA  to  A  is  —-—m, 
A 

TJ 

and  the  ratio  of  mB  to  B  is  — rj^=m, 

B 

therefore  mA  :  A  :  :  mB  :  B. 

2.  The  letter  m  still  denoting  an  integer  greater  than  unity, 
let  A  the  first  be  the  same  part  of  mA  the  second,  that  B  the 
third  is  of  mB  the  fofurth ;  then  we  are  to  show  that 

A,  mA,  B,  mB  are  proportionals. 

A        1 
The  ratio  of  A  to  mA  is  — :-= — , 
mA      m 

T>  1 

and  the  ratio  of  B  to  mB  is  ^5^=      ; 
mo      m 

therefore 

A  :  mA  :  :  B  :  mB.  Q.  E.  D. 

PROP.   D.  THEOR. 

If  the  first  be  to  the  second  as  the  third  to  the  fourth,  and 


378  APPENDIX. 

if  the  first  be  a  multiple,  or  part  of  the  second  ;  the  third  is 
the  same  multiple,  or  the  same  part  of  the  fourth. 

DEMONSTRATION. 

Any  four  proportionals  being  expressed  by 
rA,  A,  rB,  B  ; 

1.  Let  the  first  rA  be  a  multiple  of  A,  then  it  is  to  be  proved 
that  rB  is  the  same  multiple  of  B. 

Because  rA  is  a  multiple  of  A,  it  is  evident  that  r  is  an  in- 
teger greater  than  unity,  and  r  being  such  an  integer,  rA,and 
rB  are  manifestly  equimultiples  of  A  and  B. 

2.  If  r A  be  a  ^art  of  A,  w^e  are  to  show  that  rB  is  the  same 

part  of  B. 

A      1 
Because  rA  is  a  part  of  A,  therefore  — -=-  must  be  an  in- 
^  rA     r 

teger  greater  than  unity  ;  but  — ,  when  reduced,  is  also  equal 

to  ",  that  is,  to  the  same  integer,  and  therefore  rA,  rB,  are 
the  same  parts  of  A  and  B.  Q.  E.  D. 

PROP.  VII.  THEOR. 

Equal  magnitudes  have  the  same  ratio  to  the  same  magni- 
tude ;  and  the  same  has  th^  same  ratio  to  equal  magnitudes. 

DEMONSTRATION. 

Let  A  and  B  be  any  two  equal  magnitude,  and  C  any 
other,  we  arfe  to  prove  that  A  and  B  have  each  the  same  ratio 
to  C,  and  that  C  has  the  same  ratio  to  A  and  B. 

Because  by  hypothesis  A=:B, 

therefore  by  division  7^=7^-  *» 

that  is,  A  :  C  : :  B  :  C. 

Again,  since  by  hypothesis  A=B, 

C      C 
therefore  by  division  -7-= 5-  ; 
A       x5 

that  is,  C  :  A  :  :  C  :  B.  Q.  E.  D. 

PROP.  VIII.  THEOR. 

Of  unequal  magnitudes  the  greater  has  a  greater  ratio  to 
the  same,  than  the  less  has  :  and  the  same  magnitude  has  a 
greater  ratio  to  the  less,  than  it  has  to  the  greater. 

DEMONSTRATION. 

Let  A  and  B  be  two  unequal  magnitudes,  of  which  A  is  the 
greater,  and  let  C  be  any  magnitude  whatever  of  the  same 
Kind  with  A  and  B  :  it  is  to  be  shown  that  the  ratio  of  A  to  C 


APPENDIX.  379 

is  greater  than  the  ratio  of  B  to  C  :  and  also  that  the  ratio  of 
C  to  B  is  greater  than  the  ratio  of  C  to  A. 

1  Because  by  hypothesis  A>B, 

therefore,  by  division  7r>7i  5 

that  is,  the  ratio  of  A  to  C  is  greater  than  the  ratio  of  B  to  C. 

2  Because  by  hypothesis  A7B,  therefore  B^A, 

C      C 

and  therefore  by  division  we  have  ^7-jrt 

because  the  less  the  divisor  of  C  is,  the  greater  is  the  quo- 
tient ;  and  therefore  the  ratio  of  C  to  B  is  greater  than  the 
ratio  of  C  to  A.  Q.  E.  D. 

PROP.  IX.  THEOR. 

Magnitudes  which  have  the  same  ratio  to  the  same  magni- 
tude are  equal  to  one  another ;  and  those  to  which  the  sarnie 
magnitude  has  the  same  ratio,  are  equal  to  one  another. 

DEMONSTRATION. 

1.  Let  A  and  B  have  the  same  ratio  to  C,  it  is  to  be  proved 
that  A  is  equal  to  B. 

Because  A  and  B  have,  by  hypothesis,  the  same  ratio  to  C, 

A      B 

therefore  we  have  the  equality  7^=7^-1  and  therefore  by  mul- 

tiplication  A=B. 

2.  Because  by  hypothesis,  C  has  the  same  ratio  to  A  as  to 

C      C 

B,  therefore  we  have  the  equality— =.g-,  therefore,  by  divid- 

ing  by  C,  and  multiplying  by  A  and  B,  we  have  A=B. 

Q.  E.  D. 

PROP.  X.      THEOR. 

That  magnitude  which  has  a  greater  ratio  than  another  has 
to  the  same  magnitude,  is  the  greater  of  the  two :  and  that 
magnitude  to  which  the  same  has  a  greater  ratio  than  it  has  to 
another,  is  the  less  of  the  two. 

DEMONSTRATION. 

1 .  Let  A  have  to  C  a  greater  ratio  than  B  has  to  C,  it  is  to 
be  proved  that  A  is  greater  than  B. 

A  B 

Since  the  ratios  of  A  and  B  to  C,  are  7^  and  7^, 

thereftJre  by  supposition  jt^t^i  ^^^  therefore  by  mul" 

tiplication  A>B. 

2.  Here  the  ratio  of  C  to  B  is  greater  than  he  ratio  of  C 
to  A,  and  we  have  to  prove  that  B  is  less  than  to  A : 


380  APPENDIX. 

C     C 

We  have,  therefore,  by  hypothesis  -ttZ-t-. 

li     A. 

Since  then  C  contains  B  oftener  than  C  contains  A,  it  is 

manifest  that  B  must  be  less  than  A.  Q.  E.  D. 

PROP.  XI.       THEOR. 

Ratios  that  are  the  same  to  the  same  ratio,  are  the  same  to 
one  another. 

DEMONSTRATION. 

Let  A  be  to  B  as  C  to  D,  and  also  E  to  F  as  C  to  D  ;  it  is 
to  be  shown  that  A  is  to  B  as  E  is  to  F. 

A      C 

Because  A  is  to  B  as  C  to  D,  therefore,  ^^r^^yrt 

a     u 
■p      p 
for  the  same  reason  r=p=:—-;  therefore 
r       D 

^~,  that  is,  A  :  B  : :  E  :  F.  Q.  E.  D. 

D       r 

PROP.  XII 

If  any  number  of  magnitudes  be  proportionals,  as  one  of 
the  antecedents  is  to  its  consequent,  so  shall  all  the  antece- 
dents taken  together  be  to  all  the  consequents. 

DEMONSTRATION. 

By  Cor.  2.  Def.  5.  any  number  of  proportionals  may  be 
expressed  by  rA,  A  ;  rB,  B  ;  rC,  C  ; 

Where  rA,  rB,  rC,  are  the  antecedents,  and  A,  B,  C,  the 
consequents  ;  and  we  are  to  prove  that 

as  rA  is  to  A,  so  is  rA+rB+rC  to  A+B+C. 

rA 
The  ratio  of  rA  to  A  is  expressed  by  -T-=^>and  the  ratio  of 

A 

rA+rB+rC  to  A+B+C,  by  ^^^^i^^^=r  ;  and  therefore 

A-j-Jb>-f-L' 

rA  :  A  : :  rA-frB+rC  :  A+B+C. 

Q.  E.  D. 

PROP.  XIII.  THEOR. 

If  the  first  has  to  the  second  the  same  ratio  which  the  third 
has  to  the  fourth,  but  the  third  to  the  fourth  a  greater  ratio 
than  the  fifth  has  to  the  sixth  ;  the  first  shall  also  have  to  the 
second  a  greater  ratio  than  the  fifth  has  to  the  sixth. 

DEMONSTRATION, 

Let  A,  B,  C,  D,  E,  F  be  the  first,  second,  third,  fourth,  fifth, 
and  six  magnitudes  respectively. 

The  ratios  of  A  to  B,  of  C  to  D,  and  of  E  to  F 
ACE 
"®B~  ET'F' 


4 

APPENDIX.  381 

.AC 

and  since  by  hypothesis  -o=|t-» 

and  also-^>p-, 

A     E 
therefore  we  have  -u>^r-.  ^  t^  t^ 

B     r  Q.  E.D. 

Cor.  And  if  the  first  have  a  greater  ratio  to  the  second 
than  the  third  has  to  the  fourth,  but  the  third  the  same  ratio 
to  the  fourth  which  the  fifth  has  to  the  sixth  ;  it  may  be  de- 
monstrated, in  Hke  maimer,  that  t^he  first  has  a  greater  ratio  to 
the  second  than  the  fifth  has  to  the  sixth. 

PROP.  XIV.      THEOR. 

If  the  first  has  to  the  second  the  same  ratio  which  the  third 
has  to  the  fourth  ;  then,  if  the  first  be  greater  than  the  third, 
the  second  shall  be  greater  than  the  fourth  ;  if  equal,  equal, 
and  if  less,  less. 

DEMONSTRATIOISr. 

Let  rA,  A,  rB,  B,  be  any  four  proportionals. 
1.    Suppose  rA/rB, 
then  by  division     A/  B  ; 

next,  suppose  rA=::rB, 
then  by  division     A=  B  ; 

lastly,  suppose  rA  / rB, 
then  by  division     A/  B.  Q.  E.  D. 

PROP.   XV.       THEOR. 

Magnitudes  have  the  same  ratio  to  one  another  which  their 
equimultiples  have. 

DEMOrrfSTRATION. 

Let  A,  B,  be  any  two  magnitudes  of  the  same  kind  ;  and  m 
being  any  integer  greater  than  unity,  let  mA,  mB,  be  equimul- 
tiples of  A,  B  ;  it  is  to  be  proved  that 

A,  B,  mA,  mB,  are  proportionals. 

The  ratio  of  A  to  B  is  the  numerical  quotient  rjr-,  and  the 

JD 

ratio  of  mk.  to  mB  is  — ^r-,  which  is  reducible  to  -^\  therefore 
ma  J3 

the  two  ratios  .rr-,  —z^  are  equal,  and  therefore 
Jb>   mt) 

1^  A  :  B  ; :  mA  :  mB. 

PROP.   XVI.       THEOR. 

If  four  magnitudes  of  the  same  kind  be  proportionals,  they 
shall  also  be  proportionals  when  taken  alternately. 


382  APPENDIX. 

DEMONSTRATION. 

We  may  express  any  four  proportionals  by 

rA,  A,  rB,  B, 
and  we  are  to  demonstrate  that  the  four 

rA,  rB,  A,  B, 

will  also  be  proportionals. 

rk 
The  ratio  of  rA  to  rB  is  -j-,  which,  because  the  factor  r  is 

in  both  numerator  and  denominator,  is  evidently  reducible  to 

A  ^  A 

^  ;  again  the  ratio  of  the  third  A  to  the  fourth  B  is  also  -^  ; 

D  D 

therefore,  the  two  ratios,  viz.  of  r  A  to  rB,  and  of  A  to  B,  be- 
ing equal,  we  have 

rA  :  rB  : :  A  :  B. 

Q.  E.  D. 

PROP.  XVII.       THEOR. 

If  magnitudes  taken  jointly  be  proportionals,  they  shall  also 
be  proportionals  when  taken  separately  ;  that  is,  if  two  mag- 
nitudes together  have  to  one  of  them  the  same  ratio  which 
two  others  have  to  one  of  these,  the  remaining  one  of  the  first 
two  shall  have  to  the  other  the  same  ratio  which  the  remain- 
ing one  of  the  last  two  has  to  the  other  of  these. 

DEMONSTRATION. 

By  hypothesis  we  have  A  +  B  :  B  : :  C  +  D  •*  D,  and  we 

are  to  prove  that  A  :  B  : :  C  :  D. 

A-f-B     A 
Now  the  ratio  of  A+B  to  B  is  —-^-=^+\, 

D  X> 

C-l-D     C 
and  the  ratio  of  C  +  D  to  D  is  —L  -=^+1  ; 

and  since  by  hypothesis  these  two  ratios  are  equal,  therefore 

A   .  ,      C    ,  ,  ,     A       C      ,      . 

we  have  ^  +  1  =  |^  +  I,  consequently,  —  =^  ;  that  is, 

A  :  B  : :  C  :  D. 

Q.  E.  D. 

PROP.  XVIII.       THEOR. 

If  magnitudes  taken  separately  be  proportionals,  they  shall 
also  be  proportionals  when  taken  jointly  ;  that  is,  if  tl^first 
be  to  the  second  as  the  third  is  to  the  fourth,  the  first  ami  se- 
cond together  shall  be  to  the  second  as  the  third  and  fourth 
together  to  the  fourth. 


APPENDIX.  383 


DEMONSTRATION. 

By  hypothesis  we  have  A  :  B  : :  C  :  D, 
and  we  are  to  demonstrate  that  A+B:B::C-i-D:D. 
Since  the  ratio  of  A  to  B  is  the  same  with  that  of  C  to  D, 

therefore  jt-=yti 

to  each  side  of  this  equation  add  unity,  and  we  have 

A    ,  .      C  ^,     .      .     A  +  B     C  +  D 
j3-+l=^  +  l,  that  IS, -j^=--^-; 

and  therefore  A-fB  :  B  :  :  C  +  D  :  D.  Q.  E.  D. 

PROP.  XIX.      THEOR. 

If  a  whole  magnitude  be  to  a  whole,  as  a  magnitude  taken 
from  the  first  is  to  a  magnitude  taken  from  the  other,  the  re- 
mainder shall  be  to  the  remainder  as  the  whole  to  the  whole 

DEMONSTRATION. 

Let  A,  B,  be  the  two  whole  magnitudes,  and  C,  D,  the  mag- 
nitudes taken  from  them. 

So  that  by  hypothesis  A  :  B  :  :  C  :  D, 

we  are  to  prove  that  A  :  B  :  :  A  — C  :  B  — D. 

A       B 

By  Prop.  XVI.  we  have  ^=Yr-, 

.A       ^      B       ,    ^     .     A-C     B-D 
consequently  ^ —  1  =-- —  1,  that  is,  — p—  = — t=^ — ; 

By  this  last  divide  the  first  equation, 

and  the  equal  -quotients  are  - — pr=T5- — f7 
A  —  O      D — D 

and  therefore  by  mult,  and  div.  ^=^3 — f^, 

r>       B  —  U' 

thatis,  A  :  B  ::  A~C  :  B— D.  Q.  E.  D. 

ANOTHER  DEMONSTRATION. 

Since  by  hypothesis  A  :  B  : :  C  :  D, 

therefore  by  alternation,  prop.  XVI.  A  :  C  : :  B  :  D, 

and  by  division,  prop.  XVII.  A  — C  :  C  : :  B  — D  :  D, 

and  by  alternation.  A  — C:B — D::C:D, 

and  therefore  by  prop.  XL  A  — C  :  B  — D  ::  A  :  B. 

Q.  E.  D. 

ANOTHER   DEMONSTRATION. 

J?-!  .\+C,  and  B-j-D,  be  the  whole  magnitudes,  and  C,  D, 
iiifc  magnitudes  taken  away,  so  that  by  hypothesis 


384  APPENDIX. 

A+C  :  B  +  D  ::  C  :  D. 
And  we  are  to  show  that 

A+C  :  B  +  D  ::  A  :  B. 

Since  by  hypothesis  A+C  :  B  +  D  : :  C  :  D, 

therefore  by  prop.  XVI.  A  +  C:C::B  +  D:D, 

consequently  by  prop.  XVII.  A  :  C  : :  B  :  D, 

and  therefore  by  prop.  XVI.  A  :  B  : :  C  :  D, 

therefore  by  prop.  XI.  A+C  :  B  +  D  : :  A  :  B 

Q.  E.  D 

ANOTHER  DEMONSTRATION. 

Supposing  r  greater  than  unity,  let  rA,  rB,  be  the  two 
wholes,  and  A,  C  the  magnitudes  taken  away,  so  that  by  hy 
polhesis,  we  have  rA  :  rB  :  :  A  :  C  ; 

of  course  we  have -^  =  -,  or- =-,  whence  C  =  B,  and  we 

have  therefore  only  to  show  that 

rA  :  rB  ::  rA— A  :  rB— B  ; 

rA     A 
Now  the  ratio  of  rA  to  rB  is  -fr=T^ ; 

rB     B 

,    ,  /.     .       .  T^     T^    •    *'A — A     (r— 1).A 

and  the  ratio  of  rA— A  to  rB— B,  is  -jr — -—- 77-5= 

r^  —  o     (r— 1).B 

=^  and  therefore 

rA  :  rB  :  :  rA-A  :  rB-B. 

Q.E.D 

PROP.  E.     THEOR. 

If  four  magnitudes  be  proportionals,  they  are  also  propor- 
tionals by  conversion ;  that  is,  the  first  is  to  its  excess  above 
the  second  as  the  third  is  to  its  excess  above  the  fourth. 

DEMONSTRATION. 

Let  rA,  A,  rB,  B,  be  the  four  proportionals, 
we  have  to  demonstrate  that 

rA  :  rA— A  : :  rB  :  rB— B. 

r  A  r 

The  ratio  of  r A  to  r  A— A  is 


rA-A     T'-V 

and  the  ratio  of  rB  to  rB— B  is    - — rr-=: -^ 

rn  —  a      T — 1 

therefore  rA  ;  rA— A  : :  rB  :  rB— B. 

Q.E.D 


APPENDIX.  385 

PROP.  XX.  THEOR. 

If  there  be  three  magnitudes,  and  other  three,  which  taken 
two  and  two  have  the  same  ratio  ;  if  the  first  be  greater  than 
the  third,  the  fourth  will  be  greater  than  the  sixth  ;  if  equal, 
equal ;  and  if  less,  less. 

•  DEMONSTRATION. 

Let  the  three  first  magnitudes  be  A,  B,  C, 

and  the  other  three  be  D,  E,  F ; 

so  that  by  hypothesis,  A  is  to  B  as  D  to  E,  and  B  to  C  as  E 

to  F  ;  and  it  is  to  be  proved  that  if  A  be  greater  than  C,  D 

will  be  greater  than  F  ;  if  equal,  equal ;  and  if  less,  less. 

A      D 
Because  A  :  B  :  :  D  :  E,  therefore  r=r-  =  -^, 

i   E 

and  because  B  :  C  : :  E  :  F,  therefore  ^=^ :  , 

therefore  by  multiplication  of  fractions, 

AB_DE    ,      .    A    _D^ 
FG  ""EF'  ^  ^^  ^^  C    "F  ' 

from  which  it  is  evident  that  when  the  quotient  --  is  greater 

than  unity,  the  quotient  —  is  also  greater  than  unity ;  that  is, 

if  A  be  greater  than  C,  D  is  also  greater  than  F  ;  in  a  similar 
manner  it  is  shown  that  when  A  is  equal  to  C,  D  is  equal  to 
F;  and  if  less,  less.  Q.  E.  D 

PROP.  XXI.  THEOR 

If  there  be  three  magnitude^,  and  other  three,  which  have 
the  same  ratio  taken  two  and  two,  but  in  a  cross  order  ;  if  the 
first  be  greater  than  the  third,  the  fourth  shall  also  be  greater 
than  the  sixth  ;  if  equal,  equal  ;  and  if  less,  less. 

DEMOr^STRATION. 

Let  the  three  first  magnitudes  be  A,  B,  C, 
and  the  other  three  be  D,  E,  F, 
SO  that  A  is  to  B  as  E  to  F,  and  B  to  C  as  D  to  E  ;  it  is  to  be 
shown  that  if  A  be  greater  than  C,  D  will  be  greater  than  F  ; 
if  equal,  equal  ;  and  if  less,  less. 

A       F 
Since  A  :  B  : :  E  :  F,  therefore   we  have  :r— =:=rj3^^™dbe- 

rJ       r 

B       D 
cause  B  :  C  : :  D  :  E,  therefore  also  7^==-  ;  and  therefore 

O       Jb4 

by  multiplication,  3^ 


386  APPENDIX. 

AB_DE  A  __D 

from  which  it  is  manifest,  that  according   as   the   quotid  .= 
-^  is   greater  than,  equal  to,  or  less  than  unity,  the  quotient 

^  must  also  be  greater  than,  equal  to,  or  less  than  unity,  and 

therefore  if  A  be  greater  than  C,  D  will  be  greater  than  F ; 
if  equal,  equal ;  and  if  less,  less. 

PROP.  XXII.  THEOR. 

If  there  be  any  number  of  magnitudes,  and  as  many  others, 
which,  taken  two  and  two  in  order,  have  the  same  ratio ;  the 
first  shall  have  to  thegfast  of  the  first  rank  of  magnitudes,  the 
same  ratio  which  the  first  of  the  others  has  to  the  last. 

N.  B.  This  is  usually  cited  by  the  words  ex  aquali,  or  ex 
(Bquo. 

DEMONSTRATION. 

Let  the  first  rank  of  magnitudes  be  A,  B,  C,  D, 

and  the  second  rank  be  E,  F,  G,  H, 

so  that  by  hypothesis  A  is  to  B  as  E  to  F,  B  to  C  as  F  to  G, 

and  C  to  D  as  G  to  H  ;  we  are  to  show  that  A  :  D  :  :  E  :  H. 

A      E 
Since  A  :  B  :  :  E  :  F,  therefore  we  have  —  =r-^, 

o      F 

R       F 
in  like  manner  we  have  —-=:—-, 
O      G 

,  C      G 

ABC 

now  multiply  the  quotients  — ,  — ,  =-  together,  and  also  the 

E    F     G        _,        ^        ^  .      ABC     EFG 

quotients  j^,  ~,  ^,  and  we  have  the  equation  g^=:— ^, 

which  by  reduction  becomes  rp— =— -, 

JJ       rl 

and  therefore  A  :  D  :  :  E  :  H. 

In  like  manner  the  truth  of  the  proposition  may  be  shown, 

whatever  be  the  number  of  magnitudes. 

Q.  E.  D. 

PROP.  XXIII.  THEOR. 

If  there  be  any  number  of  magnitudes,  and  as  many  others 
which,  taken  two  and  two  in  a  cross  order,  have  ihe  same  ra« 


A 

G 

B 

^H' 

B 

F 

C 

~G* 

C 

E 

D 

-F' 

APPENDIX.  387 

tio  ;  the  first  shall  have  to  the  last  of  the  first  magnitudes  the 
same  ratio  which  the  first  of  the  others  has  to  the  last. 

N.  B.  This  is  usually  cited  by  the  words  ex  equali  in  pro- 
portione  perturbata,  or  ex  cequo  perturbato  ;  that  is,  by  equality 
in  per tur bate  proportion.  • 

DEMONSTRATION. 

Let  the  first  rank  of  magnitudes  be  A,  B,  G,  D, 
and  the  other  rank  E,  F,  G,  H, 
so  that,  by  hypothesis,  A  is  to  B  as  G  to  H  ;  B  to  C  as  F  to 
G,  and  C  to  D  as  E  to  F  ;  we  are  to  prove,  that 
A  :  D  : :  E  :  H. 

Since  A  :  B  : :  G  :  H,  therefore 
and  because  B  :  0  ^F  :  G,  therefore, 

and  because  C  :  D  : ;  E  :  F,  therefore 

ABC 

now  multiply  the  quotients  ^^  p-,  ^ri  together,  and  also  the 

G    F    E        ^       ^        ,  1    .    ABC      GFE 

quotients  — '  7T-»  p-,  and  we  have  the  products  uq^=  hqF* 

which  reduced,  becomes  .p-=:=j-, 
D       Jti 

and  therefore  A  :  D  : :  E  :  H. 

In  like  manner  we  may  proceed  for  any  number  of  magni- 
tudes. Q.  E.  D. 

PROP.  XXIV. 

If  the  first  has  to  the  second  the  same  ratio  which  the  third 
has  to  the  fourth  ;  and  the  fifth  to  the  second  the  same  ratio 
which  the  sixth  has  to  tke  fourth ;  the  first  and  fifth  together 
shall  have  to  the  second  the  same  ratio  which  the  third  and 
sixth  together  have  to  the  fourth.        • 

DEMONSTRATION. 

By  hypothesis  we  have  rA  :  A  : :  rB  :  B, 

and  r'A  :  A  :  :  r'B  :  B, 

in  which  rA  is  the  first,  A    the  second,  rB  the  third,  B  the 

fourth,  r'A  the  fifth,  and  r^B   the  sixth  :  r'  denoting  each  of 

the  two  equal  ratios  when  the  fifth  is  divided  by  the  second. 

and  the  sixth  by  the  fourth  ;  and  we  have  to  show,  that 

rA+r'A  :  A::rB  +  r'B  :  B. 

rA  +  r'A 
The  ratio  of  rA-i-r'A  to  A  is =r-{  /, 


388  APPENDIX. 

rB-f-r'B 
and  the  ratio  of  rB+^'B  to  B  is — =r-\-r^ ; 

D 

therefore,  rA-fr'A  :  A  : :  rB  +  r'B  :  B. 

Q.E.D 

Cor.  1.  If  the  same  hypothesis  be  made  as  in  the  propo- 
sition, the  excess  of  the  first  and  fifth  shall  be  to  the  second 
as  the  excess  of  the  third  and  sixth  to  the  fourth. 

CoR.  2.  The  prop,  holds  true  of  two  ranks  of  magnitudes, 
whatever  be  their  number,  of  which  each  of  the  first  rank 
has  to  the  second  magnitude  the  same  ratio  which  the  corres- 
ponding one  of  the  second  rank  has  to  a  fourth  magnitude. 

PROP.  XXV.      THEOR. 

If  four  magnitudes  of  the  same  kind  be  proportionals,  the 
greatest  and  least  of  ihem  together  ar^reater  than  the  other 
two  together. 

DEMONSTRATION. 

Let  the  proportionals  be  rA,  A,  rB,  B  ; 
and  let  the  iirst  rA  be  the  greatest :  then  since  by  hypothesis 
rA  is  the  greatest,  rXy  A,  therefore  r/l. 

Again,  since  by  hypothesis  rA  is  the  greatest,  therefore 
rA  7" rB,  and  consequently  A  /^ B  ;  since  then  r  is  greater  than 
unity,  and  A  is  greater  than  B,  it  is  manifest  that  B  is  the 
least ;  and  we  are  to  show  that  rA-(-B7^B-f-A 

Now  because  A  — B  — A— B, 

and        »*Z1» 
therefore,  by  multiplication         rA  — rB/^A — B  ; 
.to  each  side  of  this  equation  add  rB+B, 

and  we  shall  have  rA-fB/A  +  rB. 

A  similar  mode  of  demonstration  may  be  adopted,  which- 
ever of  the  four  proportionals  be  the  greatest. 

^  Q.  E.  D. 

PROP.  XXVf.  THEOR. 

If  there  be  any  number  of  magnitudes  of  the  same  kind, 
the  ratio  compounded  of  the  ratios  of  the  first  to  the  second, 
of  the  second  to  the  third,  and  so  on  to  the  last,  is  equal  to  the 
ratio  of  the  first  to  the  last. 

DEMONSTRATION. 

Let  the  magnitudes  of  the  same  kind  be  A,  B,  C,  D  ;  we  are 
to  prove  that  the  ratio  compounded  of  the  ratios  of  A  to  B,  of 
B  to  C,  and  of  C  to  D,  according  to  the  definition  of  com- 
pound ratio,  is  equal  to  the  ratio  of  A  to  D. 


APPENDIX.  389 

Take  any  magnitude  P, 
and  let  A  be  to  B  as  P  to  Q,  and         A,  B,  C,  D, 
B  to  C  as  Q  to  R,  and  C  to  D  as         P,  Q,  R,  S  ; 
R  to  S  ;  then  by  the  definition  of 

compound  ratio,  the  ratio  of  P  to  S  is  the  ratio  compounded  of 
the  ratios  of  A  to  B,  B  to  C,  and  of  C  to  D  ;  and  it  is  to  be 
proved  that  the  ratio  of  A  to  D  is  the  same  with  P  to  S. 

Now  because  A,  B,  C,  D,  are  several  magnitudes,  anj  P, 
Q,  R,  S,  as  many  others,  which,  taken  two  and  two.  in  order, 
have  the  same  ratio  ;  that  is,  A  is  to  B  as  P  to  Q  ;  B  to  C  as 
Q  to  R,  and  C  to  D  as  R  to  S  ;  therefore  ex  equali,  prop. 
XXII. 

A  :  D  : :  P  :  S. 

In  like  manner  the  proposition  is  proved  for  any  number  of 
magnitudes. 

Q.E.D. 

PROP.  XXVII.       THEOR. 

If  four  magnitudes  be  proportionals  according  to  the  com- 
mon algebraic  definition,  they  will  also  be  proportionals  ac- 
cording to  Euclid's  definition. 

DEMONSTRATION. 

Let  the  four  rA,  A,  rB,  B, 

be  the  proportionals  according  to  our  fifth  definition  ;  that  is, 
according  to  the  common  algebraic  definition  ;  it  is  to  be  proved 
that  the  same  four 

rA,  A,  rB,  B, 
are  proportionals  by  Euclid's  fifth  def.  of  the  fifth  book. 

Let  m  and  n  be  any  two  integer^,  each  greater  than  unity, 
so  that  mrA,  mrB,  are  aliy  equimultiples  whatever  of  the  first 
and  third  ;  and  nA,  wB,  are  any  whatever  of  the  second  and 
fourth  ;  and  the  four  multiples  are  therefore 

mrA,  nA,  mrB,  wB  ; 
Now  the  thing  to  be  proved  is,  that  according  as  the  multiple 
mrA  is  greater  than,  equal  to,  or  less  than  nA  ;  the  multiple 
mrB  will  also  be  greater  than,  equal  to,  or  less  than  nB. 

First  let  TwrA/wA,     - 
then  by  division  mryn^ 

and  by  multiplication  mrBynB. 

Secondly,  if  nirA  =  nA, 

then  mr=n, 

and  therefore  wirB=nB. 

Lastly,  if  ftrA^nA, 

then  mr/^rij 

therefore  mrB  /  nB.  Q.  E.  D. 


390  APPENDIX 

PROP.  XXVIII.      THEOR. 

If  four  magnitudes  be  proportionals  by  Euclid's  fifth  defini- 
tion, they  will  also  be  proportionals  by  the  common  algebraic 
definition. 

DEMONSTRATION. 

Let  A',  A,  B',  B,  be  any  four  magnitudes,  such  that  m,  n, 

being  any  integers  greater  than  unity,  and  the  equimultiples, 

mA^,  mB\  being  taken,  and  likewise  the  equimultiples  nA,  nB  ; 

making  the  four  multipyes 

mA',  nA,  mB',  nB  ; 

the  hypothesis  is,  that  if  twA'  be  greater  than  nA,  mB' is  also 

greater  than  nB  ;  if  equal,  equal ;  and  if  less,  less  :  and  it  is 

to  be  proved  that 

A'  :  A  : :  B'  :  B  ; 

A''     B'' 
or,  which  is  the  same  thing,  that  —  =-p-. 

A'  B' 

If  —  be  not  equal  to  ^g- ,  one  of  these  quolients  must  be  the 

A'  B' 

greater ;  first,  let  —  be  the  greater,  so  that  if  :5-=:r,  we  may 

A' 
have  j-=:zr+r' ; 

then  the  four  quantities  A',  A,  B',  B, 
are  equal  to  rA  +  rA,  A,  rB,  B. 

Now,  let  m  be  such  an  integer  greater  than  unity,  that  mr 
and  mr'  may  be  each  greater  than  2  ;  and  take  n  the  next  in- 
teger greater  than  mr,  of  course  n  will  be  less  than  mr~{-mr' ; 
and  the  four  multiples  mA',  nA,  mB',  nB, 
become  mrA+mr'A,  nA,  mrB,  nB, 

By  construction  mr-\-mr'yn, 

and  therefore  mr  A  -f  mr' A  7  nk  ; 

But  by  construction  mr<:^n, 

and  therefore  mrB<nB; 

or  mB'<nB  ; 

thus  it  appears  that  m\"/n\, 

but  mB'<«B  : 

but,  because  mA'>nA, 

therefore,  by  hypothesis,  also  mh^ynB  ; 

so  that  mB'  is  both  greater  and  less  than  nB,  which  is  impos- 
sible # 

A'  B' 

It  is  manifest  therefore  that  -j-  cannot  be  greater  than  fr-  ; 

A  ii 


APPENDIX.  391 

and  in  like  manner  it  is  shown  that  jr  cannot  be  greater  than 

■J-;  and  tnereCore  — =p-, 

that  is  A'  :  A  . :  B'  :  B.  Q.  E.  D. 

Scholium.  Thus  we  have  shown,  that  if  four  quantities  be 
proportionals  by  the  common  algebraic  definition,  they  will 
also  be  proportionals  according  to  Euclid's  definition  ;  and  con- 
versely, that  if  four  quantities  be  proportionals  by  Euclid's  de- 
finition, they  will  also  be  proportionals  by  the  common  algebraic 
definition  ;  and  by  a  similar  method  of  reasoning  we  may  easily 
show,  that  when  four  quantities  are  not  proportionals  by  one  of 
these  two  definitions,  they  cannot  be  proportionals  by  the  other 
definition. 

Thus  it  appears,  that  the  two  definitions  are  altogether: 
equivalent ;  each  comprehending,  or  excluding,  whatever  is 
comprehended,  or  excluded,  by  the  other. 


THE    END 


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